If an operation can be performed in m different ways, following which another operation can be performed in n possible ways and the operations are independent, then both operations in succession can be performed exactly in

How many different numbers of three distinct digits can be formed with the digits 0, 1, 2, 3, 4 and 5 which are divisible by 5?

Ten students compete in a race. In how many ways can the first three places be taken?

A football stadium has four entrance gates and nine exits. In how many different ways can a man enter and leave the stadium?

There are six doors in a hostel. In how many ways can a student enter the hostel and leave by a different door?

In how many ways can a man send three of his children to seven different colleges of a certain town?

Suppose there are five main roads between the cities A and B. In how many ways can a man go from a city to the other and return by a different road?

There are five main roads between the cities A and B and 4 between B and C. In how many ways can a person drive from A to C and return without driving on the same road twice?

How many numbers of at least three different digits can be formed from the integers $\rm 1, 2, 3, 4, 5, 6$ ?

How many numbers of three digits less than 500 can be formed from the integers  $\rm 1, 2, 3, 4 , 5, 6\; $?

Of the numbers formed by using all the figures$\rm 1, 2, 3, 4, 5 $ only once, how many are even?

How many numbers between 4000 and 5000 can be formed with the digits $\rm 2, 3, 4 ,5, 6, 7$ ?

How many numbers of three digits can be formed from the integers $\rm 2, 3, 4, 5, 6 $ ? How many of them will be divisible by 5 ?

Find the number of permutations of five different objects taken three at a time.

If three persons enter a bus in which there are ten vacant seats, find in how many ways they can sit.

How many plates of vehicles consisting of 4 different digits can be made out of the integers $\rm 4, 5, 6, 7, 8, 9 \;$? How many of these numbers are divisible by 2?

How many numbers of 4 different digits can be formed from the digits $\rm 2, 3, 4, 5, 6, 7 $ ? How many of these numbers are $ i) $ divisible by 5$\;$ $ ii)$  not divisible by 5. 

How many 5-digit odd numbers can be formed using the digits $\rm 3, 4, 5, 6, 7, 8,$ and $\rm 9 $. If $\rm i)$ repetition of digits is not allowed $\;$ $\rm ii)$ repetition of digits is allowed?

In how many ways can four boys and three girls be seated in a row containing seven seats 

1. if they may sit anywhere
2. if the boys and girls must alternate
3. if all three girls are together
4. if girls are to occupy odd seats

In how many ways can eight people be seated in a row of eight seats so that two particular persons are $\rm a)$ always together $\rm b) $ never together? 

Six different books are arranged on a shelf. Find the number of different ways in which the two particular books are $\rm a) $ always together $\rm b)$ not together. 

In how many ways can four red beads, five white beads, and three blue beads be arranged in a row?

In how many ways can the letters of the following words be arranged?

1. ELEMENT
2. NOTATION
3. MATHEMATICS
4. MISSISSIPPI

How many numbers of 6 digits can be formed with the digits $ 2, 3, 2, 0, 3, 3\;$?

In how many ways can 4 Art students and 4 science students be arranged in a circular table if $\rm a) $ they may sit anywhere $\rm b) $ they sit alternately? 

In how many ways can eight people be seated at a round table if two people insist on sitting next to each other?

In how many ways can seven different coloured beads be made into a bracelet?

In how many ways can 4 letters be posted in six-letter boxes?

In how many ways can the letters of the word “MONDAY” be arranged? How many of these arrangements do not begin with M? How many begin with M and do not end with Y?

Show that the number of ways in which the letters of the word “COLLEGE” can be arranged so that the two E's always come together is 360.

In how many ways can the letters of the word ‘COMPUTER’ be arranged so that 

1. all the vowels are always together?
2. the vowels may occupy only odd positions?
3. the relative positions of vowels and consonants are not changed?

Find the number of arrangements of the letters of the word “LAPTOP” so that 

1. the vowels may never be separated;
2. all the consonants may not be together;
3. they always begin with L and end with T
4. They do not begin with L but always end with T.

How many different words can be formed with all the letters of the word “INTERNET” if

1. each word is, to begin with vowel?
2. each word is to end with consonant?

How many even numbers of 3 digits can be formed when repetition of digits is allowed?

In how many ways can 3 prizes be distributed among 4 students so that each student may receive any number of prizes?

Show that the number of ways in which the letters of the word “ARRANGE” can be arranged so that no two R's come together is 900.

A boy puts his hand into a bag which contains 10 different coloured marbles and brings out 3. How many different results are possible? 

Find the number of ways in which a student can select 5 courses out of 8 courses. If 3 courses are compulsory, in how many ways can the selections be made?

From 10 persons, in how many ways can a selection of 4 be made

1. When one particular person is always included?
2. When two particular persons are always excluded?

A bag contains 8 white balls and 5 blue balls. In how many ways can 5 white balls and 3 blue balls be drawn?

How many committees can be formed from a set of 7 boys and 5 girls if each committee contains 4 boys and 3 girls?

From a group of 11 men and 8 women, how many committees consisting of 3 men and 2 women are possible?

From 4 mathematicians, 6 statisticians, and 5 economists, how many committees of 6 members can be formed so as to include 2 members from each category?

A person has got 12 acquaintances of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them may be relatives?

There are ten electric bulbs in the stock of a shop out of which there are three defectives. In how many ways can a selection of 6 bulbs be made so that 4 of them may be good bulbs?

From 6 gentlemen and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done so as to include at least one lady?

A candidate is required to answer 6 out of 10 questions which are divided into 2 groups, each containing 5 questions and he is not permitted to attempt more than 4 from any group. In how many different ways can he make up his choice?

A man has 5 friends. In how many ways can he invite one or more of them to a dinner?

If C(20, r+5) = C (20, 2r - 7), find C(15, r).

If $ \rm C(n, 10) + C(n, 9) = C(20, 10) $ find $\rm n$ and $\rm C(n, 17)$

Solve for $\rm n$ the equation $C(n+2, 4) = 6C(n, 2)$

If $\rm P(n, r) = 336$ and $\rm C(n, r) = 56$, find $\rm n$ and $\rm r$.

If $\rm ^{n} C_{r-1} = 45, ^{n}C_r = 120$ and $\rm ^nC_{r+1} = 210$ find $\rm n$ and $\rm r$.

An examination paper consisting of 10 questions, is divided into two groups A and B. Group A contains 6 questions. In how many ways can an examinee attempt 7 questions 

1. selecting 4 from group A and 3 from group B?
2. Selecting at least two questions from each group?

Six men in a group of 8 are skilled. Find the number of ways by which 5 men can be selected such that

1. at least 3 of them may be the skilled men.
2. at least one of them may be the unskilled man. 

In a group of 10 students, 6 are boys. In how many ways can 4 students be selected for mathematical competition so as to include 

1. exactly two boys
2. at least two boys
3. at most two girls.

Expand by binomial theorem and simplify.

$\rm (a + b)^7 $

Expand by binomial theorem and simplify.

$\rm (2x - 3y)^4 $

Expand by binomial theorem and simplify.

$\rm (2x + y^2)^5 $

Expand by binomial theorem and simplify.

$\rm (\frac{x}{2} + \frac{2}{y})^5 $

Expand by binomial theorem and simplify.

$\rm (x^2 + \frac {2}{y})^{5} $

Find seven term of $\rm (2x^2 + \frac {1}{x})^8 $.

Find seven term of $\rm (2x + y)^{12} $

Find seven term of $\rm (x - \frac {2}{x})^7 $

Find the general term of $\rm (x^2 + \frac {1}{x})^6 $

Find the general term of $(\rm\frac {a}{b} + \frac {b}{a})^{2n + 1} $

Find the coefficient of $\rm x^5 $ in the expansion of $(\rm x + \frac {1}{2x})^7 $

Find the coefficient of $\rm x^2 $ in the expansion of $(\rm x^3 + \frac {a}{x})^{10} $

Find the coefficient of $\rm x^6 $ in the expansion of $(\rm 3x^2 - \frac {1}{3x})^9 $

Find the term independent (free) of $\rm x$ in the expansion of $(\rm x^2 + \frac {1}{x})^{12} $.

Find the term independent (free) of $\rm x$ in the expansion of $(\rm 2x + \frac {1}{3x^2})^9$.

Find the term independent (free) of $\rm x$ in the expansion of $ (\rm x + \frac {1}{x})^{2n} $.

Find the term independent (free) of $\rm x$ in the expansion of $(\rm \frac {3x^2}{2} - \frac {1}{3x})^9 $.

Write down the fourth term in the expansion of $(\rm px + \frac {1}{x})^n$. If this term is independent of $\rm x$, find the value of $\rm n$. With this value of n, calculate the value of $\rm p $ given that the fourth term is $\rm \frac {5}{2}$.

If the coefficient of $\rm x^{-1}$ in the expansion of $(\rm x + \frac {k}{x^2})^5$ is 90, find the value of k. 

Find the middle term or terms in the expansion of $(\rm x + \frac {1}{x})^{18} $.

Find the middle term or terms in the expansion of $(\rm 2x + \frac {1}{x})^{17} $.

Find the middle term or terms in the expansion of $(\rm \frac {x}{a} - \frac {a}{x})^{2n + 1} $.

Find the middle term or terms in the expansion of $(\rm ax - \frac {1}{ax})^{2n} $.

Show that the middle term of the expansion of $\rm (1 + x )^{2n} \;$ is $\;\rm \frac{1. 3. 5.\dots .(2n - 1)}{n!}2^n x^n$

Show that the middle term of the expansion of $\rm (x -\frac{1}{x} )^{2n}\; $ is $\;\rm \frac{1. 3. 5.\dots .(2n - 1)}{n!}(-2)^n $

In the expansion of $\rm (1 + x)^{21}$, the coefficient of $\rm (2r + 1)$th term is equal to the coefficient of $\rm (3r + 1)$th term. Find r. 

In the expansion of $\rm (1 + x)^{2n+1} $, the coefficients of $\rm x^r$ and $\rm x^{r + 1}$ are equal. Find r.

Show that the coefficients of the middle term of $\rm ( 1 + x)^{2n} $ is equal to the sum of the coefficients of the two middle terms of $\rm ( 1 + x)^{2n -1}$.

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_1 - 2C_2 + 3C_3 - \dots + n ( -1 )^{n - 1]. C_n = 0$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 2C_1 + 3C_2 + \dots + (n + 1 ) C_n = ( n + 2 ) 2^{n - 1}$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 3C_1 + 5C_2 + \dots + (2n + 1 ) C_n = ( n + 1) 2^{n-1}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 4C_1 + 7C_2 + 10C_3 + \dots + ( 3n + 1 ) C_n = (3n + 2).2^{n - 1}$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm \frac {C_1}{C_0} + \frac {2.C_2}{C_1} + \frac {3.C_3}{C_2} +\dots + \frac {n C_n}{C_{n - 1} = \frac {n(n + 1)}{2}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0C_1+ C_1C_{n - 1}+ \dots + C_nC_0 = \frac {2n!}{n!\; n!}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm {C_0}^2 + {C_1}^2 + {C_2}^2 + \dots + {C_n}^2 = \frac {2n!}{(n!)^2}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0C_1 + C_1C_2 + C_2C_3 + \dots + C_{n - 1}.C_n = \frac {2n!}{( n + 1)! \;( n - 1)!}$

If the three consecutive coefficients in the expansion of $\rm ( 1 + x)^n $ be 165, 330, 462; find n.

If the four consecutive coefficients in the expression of $\rm ( 1 + x )^n $ be $\rm a_1, a_2, a_3,$ and $\; a_4$, then prove that $\rm \frac {a_1}{a_1 + a_2} + \frac {a_3}{a_3 + a_4} = \frac {2a_2}{a_2 + a_3}$

Show that

$\rm ( 1 + \frac {1}{1!} + \frac {1}{2!} + \frac {1}{3!} + \dots )( 1 - \frac {1}{1!} + \frac {1}{2!} - \frac {1}{3!}+ \dots ) = 1 $

Show that

$\rm ( 1 + \frac {1}{2!} + \frac {1}{4!} + \dots )^2 - ( 1 + \frac {1}{3!} + \frac {1}{5!} + \dots )^2 = 1 $

Show that

$\rm\frac {2}{1!} + \frac {4}{3!} + \frac {6}{5!} + \dots $ to $\rm \infty = e $

Show that

$\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 3}{3!} + \frac {1 + 2 + 3+ 4}{4!} + \dots = \frac {3e}{2} $

Show that 

$\frac {1 + \frac{1}{2!} + \frac {1}{4!} + \frac {1}{6!} + \dots }{1 + \frac{1}{3!} + \frac {1}{5!} + \frac {1}{7!} + \dots } = \frac {e^2 + 1}{e^2 - 1}$ 

Show that

$\rm \frac {1}{1!} + \frac {1 + 3}{2!} + \frac {1 + 3 + 5}{3!} + \frac {1 + 3 + 5 + 7}{4!} + \dots = 2e $

Sum to infinity 

$\rm \frac {1.2}{1!} + \frac {2.3}{2!} + \frac {3.4}{3!} + \dots $

Sum to infinity 

$\rm 1 + \frac {3}{1!} + \frac {5}{2!} + \frac {7}{3!} + \dots $

Sum to infinity 

$(\rm 1 + \frac {1}{1.2} + \frac {1}{1.2.3} + \dots )(\rm 1 - \frac {1}{1.2} + \frac {1}{1.2.3} - \dots )$

Sum the following into infinity

$(\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 2^2}{3!} + \dots )$

Show that:

 $\sum_{n=1}^\infty \frac {n^2}{(n + 1 )!} = e - 1$.

Show that:

 $\sum_{n=1}^\infty \frac {n^2}{(n - 1 )!} = 5e$.

Prove that :

$\rm \frac {1}{2.3} + \frac {1}{4.5} + \frac {1}{6.7} + \dots = 1 - \log_e2 $

Prove that :

$\rm \frac {1}{2} - \frac {1}{2.2^2} + \frac {1}{3.2^3} - \frac {1}{4.2^4} + \dots =  \log\frac {3}{2} $

Prove that :

$\rm (\frac {1}{3} - \frac {1}{2}) + \frac {1}{2}(\frac {1}{3^2} + \frac {1}{2^2}) + \frac {1}{3}(\frac {1}{3^3} - \frac {1}{2^3}) + \dots = 0 $

Prove that :

$\rm \frac {1}{n + 1} + \frac {1}{2 ( n + 1)^2} + \frac {1}{3 ( n + 1)^3} + \dots = \frac {1}{n} - \frac {1}{2n^2} + \frac {1}{3n^2} - \dots $

prove that :

$\rm 1 + \frac {1}{3.2^2} + \frac {1}{5.2^4} + \frac {1}{7.2^6} + \dots = \log_e3$

If $\rm y = x + \frac {1}{2}x^2 + \frac {1}{3}x^3 + \frac {1}{4}x^4 + \dots $, show that $\rm x = y - \frac {1}{2!}y^2 + \frac {1}{3!}y^3 - \frac {1}{4!}y^4 + \dots $

If $\rm y = \frac {x}{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + \dots $ to $\infty $ , prove that $\rm x = y - \frac {y^2}{2} + \frac {y^3}{3} - \frac {y^4}{4} + \dots $ to $\infty $ 

If $\rm x = \frac {y}{1!} - \frac {y^2}{2!} + \frac {y^3}{3!} - \frac {y^4}{4!} + \dots $, show that $\rm y = x + \frac {x^2}{2} + \frac {x^3}{3} + \frac {x^4}{4} + \dots $

If (n + 2)! = 210(n- 1)!, then the value of n satisfying the condition is

The number of ways that 8 beads of different colors be string as a neckless is

In how many different ways can 9 people and a host be seated in a circular table of a party?

In how many ways 5 boys and 5 girls sit on a circle so that no two boys sit together?

If a polygon has 44 diagonals, then the number of its sides are:

The number of arrangement of r things out of n number of identical things is

The number of ways to fill the r^th position out of n distinct things in a row is

The value of n, when p(n, 6) =3.p(n, 5) is

The letters of the word SERIES are arranged at random. How many of these arrangements has E's together?

What is the number of permutations of letter of word DAUGHTER. so that vowels occupying even places?

The number of words which can be formed from the letters of the word MAXIMUM, if two consonant cannot occur together is

how many different ways the 7 different colored beads can be strung on a necklace?

If p(n, r) = C(n, r), then

The value of r, when C(8, r) - C(7, 3) = C(7, 2) is

In a cricket tournament, the total numbers of match played is 153. How many teams were there if every team played a match with one another?

What is the number of different sums of money can be made from the 4 coins of different denominator?

Value of P(n, n) is equal to

How many groups, each of 2 vowels and 3 consonants can be formed from the letter of the word COMPUTER?

In how many ways can 5 boys and 5 girls sit on a circle so that no two boys sit together

A person has 4 friends. In how many ways can he in invited one or more friends for a party?

If P(n, r) = 360 and C(n, r) = 15. What is the value of r?

How many committees of 5 members can be formed from 6 gentlemen and 4 ladies?

If an operation can be performed in m different ways, following which another operation can be performed in n possible ways and the operations are independent, then both operations in succession can be performed exactly in

How many different numbers of three distinct digits can be formed with the digits 0, 1, 2, 3, 4 and 5 which are divisible by 5?

Ten students compete in a race. In how many ways can the first three places be taken?

A football stadium has four entrance gates and nine exits. In how many different ways can a man enter and leave the stadium?

There are six doors in a hostel. In how many ways can a student enter the hostel and leave by a different door?

In how many ways can a man send three of his children to seven different colleges of a certain town?

Suppose there are five main roads between the cities A and B. In how many ways can a man go from a city to the other and return by a different road?

There are five main roads between the cities A and B and 4 between B and C. In how many ways can a person drive from A to C and return without driving on the same road twice?

How many numbers of at least three different digits can be formed from the integers $\rm 1, 2, 3, 4, 5, 6$ ?

How many numbers of three digits less than 500 can be formed from the integers  $\rm 1, 2, 3, 4 , 5, 6\; $?

Of the numbers formed by using all the figures$\rm 1, 2, 3, 4, 5 $ only once, how many are even?

How many numbers between 4000 and 5000 can be formed with the digits $\rm 2, 3, 4 ,5, 6, 7$ ?

How many numbers of three digits can be formed from the integers $\rm 2, 3, 4, 5, 6 $ ? How many of them will be divisible by 5 ?

Find the number of permutations of five different objects taken three at a time.

If three persons enter a bus in which there are ten vacant seats, find in how many ways they can sit.

How many plates of vehicles consisting of 4 different digits can be made out of the integers $\rm 4, 5, 6, 7, 8, 9 \;$? How many of these numbers are divisible by 2?

How many numbers of 4 different digits can be formed from the digits $\rm 2, 3, 4, 5, 6, 7 $ ? How many of these numbers are $ i) $ divisible by 5$\;$ $ ii)$  not divisible by 5. 

How many 5-digit odd numbers can be formed using the digits $\rm 3, 4, 5, 6, 7, 8,$ and $\rm 9 $. If $\rm i)$ repetition of digits is not allowed $\;$ $\rm ii)$ repetition of digits is allowed?

In how many ways can four boys and three girls be seated in a row containing seven seats 

1. if they may sit anywhere
2. if the boys and girls must alternate
3. if all three girls are together
4. if girls are to occupy odd seats

In how many ways can eight people be seated in a row of eight seats so that two particular persons are $\rm a)$ always together $\rm b) $ never together? 

Six different books are arranged on a shelf. Find the number of different ways in which the two particular books are $\rm a) $ always together $\rm b)$ not together. 

In how many ways can four red beads, five white beads, and three blue beads be arranged in a row?

In how many ways can the letters of the following words be arranged?

1. ELEMENT
2. NOTATION
3. MATHEMATICS
4. MISSISSIPPI

How many numbers of 6 digits can be formed with the digits $ 2, 3, 2, 0, 3, 3\;$?

In how many ways can 4 Art students and 4 science students be arranged in a circular table if $\rm a) $ they may sit anywhere $\rm b) $ they sit alternately? 

In how many ways can eight people be seated at a round table if two people insist on sitting next to each other?

In how many ways can seven different coloured beads be made into a bracelet?

In how many ways can 4 letters be posted in six-letter boxes?

In how many ways can the letters of the word “MONDAY” be arranged? How many of these arrangements do not begin with M? How many begin with M and do not end with Y?

Show that the number of ways in which the letters of the word “COLLEGE” can be arranged so that the two E's always come together is 360.

In how many ways can the letters of the word ‘COMPUTER’ be arranged so that 

1. all the vowels are always together?
2. the vowels may occupy only odd positions?
3. the relative positions of vowels and consonants are not changed?

Find the number of arrangements of the letters of the word “LAPTOP” so that 

1. the vowels may never be separated;
2. all the consonants may not be together;
3. they always begin with L and end with T
4. They do not begin with L but always end with T.

How many different words can be formed with all the letters of the word “INTERNET” if

1. each word is, to begin with vowel?
2. each word is to end with consonant?

How many even numbers of 3 digits can be formed when repetition of digits is allowed?

In how many ways can 3 prizes be distributed among 4 students so that each student may receive any number of prizes?

Show that the number of ways in which the letters of the word “ARRANGE” can be arranged so that no two R's come together is 900.

A boy puts his hand into a bag which contains 10 different coloured marbles and brings out 3. How many different results are possible? 

Find the number of ways in which a student can select 5 courses out of 8 courses. If 3 courses are compulsory, in how many ways can the selections be made?

From 10 persons, in how many ways can a selection of 4 be made

1. When one particular person is always included?
2. When two particular persons are always excluded?

A bag contains 8 white balls and 5 blue balls. In how many ways can 5 white balls and 3 blue balls be drawn?

How many committees can be formed from a set of 7 boys and 5 girls if each committee contains 4 boys and 3 girls?

From a group of 11 men and 8 women, how many committees consisting of 3 men and 2 women are possible?

From 4 mathematicians, 6 statisticians, and 5 economists, how many committees of 6 members can be formed so as to include 2 members from each category?

A person has got 12 acquaintances of whom 8 are relatives. In how many ways can he invite 7 guests so that 5 of them may be relatives?

There are ten electric bulbs in the stock of a shop out of which there are three defectives. In how many ways can a selection of 6 bulbs be made so that 4 of them may be good bulbs?

From 6 gentlemen and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done so as to include at least one lady?

A candidate is required to answer 6 out of 10 questions which are divided into 2 groups, each containing 5 questions and he is not permitted to attempt more than 4 from any group. In how many different ways can he make up his choice?

A man has 5 friends. In how many ways can he invite one or more of them to a dinner?

If C(20, r+5) = C (20, 2r - 7), find C(15, r).

If $ \rm C(n, 10) + C(n, 9) = C(20, 10) $ find $\rm n$ and $\rm C(n, 17)$

Solve for $\rm n$ the equation $C(n+2, 4) = 6C(n, 2)$

If $\rm P(n, r) = 336$ and $\rm C(n, r) = 56$, find $\rm n$ and $\rm r$.

If $\rm ^{n} C_{r-1} = 45, ^{n}C_r = 120$ and $\rm ^nC_{r+1} = 210$ find $\rm n$ and $\rm r$.

An examination paper consisting of 10 questions, is divided into two groups A and B. Group A contains 6 questions. In how many ways can an examinee attempt 7 questions 

1. selecting 4 from group A and 3 from group B?
2. Selecting at least two questions from each group?

Six men in a group of 8 are skilled. Find the number of ways by which 5 men can be selected such that

1. at least 3 of them may be the skilled men.
2. at least one of them may be the unskilled man. 

In a group of 10 students, 6 are boys. In how many ways can 4 students be selected for mathematical competition so as to include 

1. exactly two boys
2. at least two boys
3. at most two girls.

Expand by binomial theorem and simplify.

$\rm (a + b)^7 $

Expand by binomial theorem and simplify.

$\rm (2x - 3y)^4 $

Expand by binomial theorem and simplify.

$\rm (2x + y^2)^5 $

Expand by binomial theorem and simplify.

$\rm (\frac{x}{2} + \frac{2}{y})^5 $

Expand by binomial theorem and simplify.

$\rm (x^2 + \frac {2}{y})^{5} $

Find seven term of $\rm (2x^2 + \frac {1}{x})^8 $.

Find seven term of $\rm (2x + y)^{12} $

Find seven term of $\rm (x - \frac {2}{x})^7 $

Find the general term of $\rm (x^2 + \frac {1}{x})^6 $

Find the general term of $(\rm\frac {a}{b} + \frac {b}{a})^{2n + 1} $

Find the coefficient of $\rm x^5 $ in the expansion of $(\rm x + \frac {1}{2x})^7 $

Find the coefficient of $\rm x^2 $ in the expansion of $(\rm x^3 + \frac {a}{x})^{10} $

Find the coefficient of $\rm x^6 $ in the expansion of $(\rm 3x^2 - \frac {1}{3x})^9 $

Find the term independent (free) of $\rm x$ in the expansion of $(\rm x^2 + \frac {1}{x})^{12} $.

Find the term independent (free) of $\rm x$ in the expansion of $(\rm 2x + \frac {1}{3x^2})^9$.

Find the term independent (free) of $\rm x$ in the expansion of $ (\rm x + \frac {1}{x})^{2n} $.

Find the term independent (free) of $\rm x$ in the expansion of $(\rm \frac {3x^2}{2} - \frac {1}{3x})^9 $.

Write down the fourth term in the expansion of $(\rm px + \frac {1}{x})^n$. If this term is independent of $\rm x$, find the value of $\rm n$. With this value of n, calculate the value of $\rm p $ given that the fourth term is $\rm \frac {5}{2}$.

If the coefficient of $\rm x^{-1}$ in the expansion of $(\rm x + \frac {k}{x^2})^5$ is 90, find the value of k. 

Find the middle term or terms in the expansion of $(\rm x + \frac {1}{x})^{18} $.

Find the middle term or terms in the expansion of $(\rm 2x + \frac {1}{x})^{17} $.

Find the middle term or terms in the expansion of $(\rm \frac {x}{a} - \frac {a}{x})^{2n + 1} $.

Find the middle term or terms in the expansion of $(\rm ax - \frac {1}{ax})^{2n} $.

Show that the middle term of the expansion of $\rm (1 + x )^{2n} \;$ is $\;\rm \frac{1. 3. 5.\dots .(2n - 1)}{n!}2^n x^n$

Show that the middle term of the expansion of $\rm (x -\frac{1}{x} )^{2n}\; $ is $\;\rm \frac{1. 3. 5.\dots .(2n - 1)}{n!}(-2)^n $

In the expansion of $\rm (1 + x)^{21}$, the coefficient of $\rm (2r + 1)$th term is equal to the coefficient of $\rm (3r + 1)$th term. Find r. 

In the expansion of $\rm (1 + x)^{2n+1} $, the coefficients of $\rm x^r$ and $\rm x^{r + 1}$ are equal. Find r.

Show that the coefficients of the middle term of $\rm ( 1 + x)^{2n} $ is equal to the sum of the coefficients of the two middle terms of $\rm ( 1 + x)^{2n -1}$.

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_1 - 2C_2 + 3C_3 - \dots + n ( -1 )^{n - 1]. C_n = 0$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 2C_1 + 3C_2 + \dots + (n + 1 ) C_n = ( n + 2 ) 2^{n - 1}$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 3C_1 + 5C_2 + \dots + (2n + 1 ) C_n = ( n + 1) 2^{n-1}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0 + 4C_1 + 7C_2 + 10C_3 + \dots + ( 3n + 1 ) C_n = (3n + 2).2^{n - 1}$

If $(\rm 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm \frac {C_1}{C_0} + \frac {2.C_2}{C_1} + \frac {3.C_3}{C_2} +\dots + \frac {n C_n}{C_{n - 1} = \frac {n(n + 1)}{2}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0C_1+ C_1C_{n - 1}+ \dots + C_nC_0 = \frac {2n!}{n!\; n!}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm {C_0}^2 + {C_1}^2 + {C_2}^2 + \dots + {C_n}^2 = \frac {2n!}{(n!)^2}$

If $\rm ( 1 + x )^n = C_0 + C_1x  + C_2x^2 + \dots +C_nx^n $, prove that 

$\rm C_0C_1 + C_1C_2 + C_2C_3 + \dots + C_{n - 1}.C_n = \frac {2n!}{( n + 1)! \;( n - 1)!}$

If the three consecutive coefficients in the expansion of $\rm ( 1 + x)^n $ be 165, 330, 462; find n.

If the four consecutive coefficients in the expression of $\rm ( 1 + x )^n $ be $\rm a_1, a_2, a_3,$ and $\; a_4$, then prove that $\rm \frac {a_1}{a_1 + a_2} + \frac {a_3}{a_3 + a_4} = \frac {2a_2}{a_2 + a_3}$

Show that

$\rm ( 1 + \frac {1}{1!} + \frac {1}{2!} + \frac {1}{3!} + \dots )( 1 - \frac {1}{1!} + \frac {1}{2!} - \frac {1}{3!}+ \dots ) = 1 $

Show that

$\rm ( 1 + \frac {1}{2!} + \frac {1}{4!} + \dots )^2 - ( 1 + \frac {1}{3!} + \frac {1}{5!} + \dots )^2 = 1 $

Show that

$\rm\frac {2}{1!} + \frac {4}{3!} + \frac {6}{5!} + \dots $ to $\rm \infty = e $

Show that

$\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 3}{3!} + \frac {1 + 2 + 3+ 4}{4!} + \dots = \frac {3e}{2} $

Show that 

$\frac {1 + \frac{1}{2!} + \frac {1}{4!} + \frac {1}{6!} + \dots }{1 + \frac{1}{3!} + \frac {1}{5!} + \frac {1}{7!} + \dots } = \frac {e^2 + 1}{e^2 - 1}$ 

Show that

$\rm \frac {1}{1!} + \frac {1 + 3}{2!} + \frac {1 + 3 + 5}{3!} + \frac {1 + 3 + 5 + 7}{4!} + \dots = 2e $

Sum to infinity 

$\rm \frac {1.2}{1!} + \frac {2.3}{2!} + \frac {3.4}{3!} + \dots $

Sum to infinity 

$\rm 1 + \frac {3}{1!} + \frac {5}{2!} + \frac {7}{3!} + \dots $

Sum to infinity 

$(\rm 1 + \frac {1}{1.2} + \frac {1}{1.2.3} + \dots )(\rm 1 - \frac {1}{1.2} + \frac {1}{1.2.3} - \dots )$

Sum the following into infinity

$(\rm 1 + \frac {1 + 2}{2!} + \frac {1 + 2 + 2^2}{3!} + \dots )$

Show that:

 $\sum_{n=1}^\infty \frac {n^2}{(n + 1 )!} = e - 1$.

Show that:

 $\sum_{n=1}^\infty \frac {n^2}{(n - 1 )!} = 5e$.

Prove that :

$\rm \frac {1}{2.3} + \frac {1}{4.5} + \frac {1}{6.7} + \dots = 1 - \log_e2 $

Prove that :

$\rm \frac {1}{2} - \frac {1}{2.2^2} + \frac {1}{3.2^3} - \frac {1}{4.2^4} + \dots =  \log\frac {3}{2} $

Prove that :

$\rm (\frac {1}{3} - \frac {1}{2}) + \frac {1}{2}(\frac {1}{3^2} + \frac {1}{2^2}) + \frac {1}{3}(\frac {1}{3^3} - \frac {1}{2^3}) + \dots = 0 $

Prove that :

$\rm \frac {1}{n + 1} + \frac {1}{2 ( n + 1)^2} + \frac {1}{3 ( n + 1)^3} + \dots = \frac {1}{n} - \frac {1}{2n^2} + \frac {1}{3n^2} - \dots $

prove that :

$\rm 1 + \frac {1}{3.2^2} + \frac {1}{5.2^4} + \frac {1}{7.2^6} + \dots = \log_e3$

If $\rm y = x + \frac {1}{2}x^2 + \frac {1}{3}x^3 + \frac {1}{4}x^4 + \dots $, show that $\rm x = y - \frac {1}{2!}y^2 + \frac {1}{3!}y^3 - \frac {1}{4!}y^4 + \dots $

If $\rm y = \frac {x}{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + \dots $ to $\infty $ , prove that $\rm x = y - \frac {y^2}{2} + \frac {y^3}{3} - \frac {y^4}{4} + \dots $ to $\infty $ 

If $\rm x = \frac {y}{1!} - \frac {y^2}{2!} + \frac {y^3}{3!} - \frac {y^4}{4!} + \dots $, show that $\rm y = x + \frac {x^2}{2} + \frac {x^3}{3} + \frac {x^4}{4} + \dots $

If (n + 2)! = 210(n- 1)!, then the value of n satisfying the condition is

The number of ways that 8 beads of different colors be string as a neckless is

In how many different ways can 9 people and a host be seated in a circular table of a party?

In how many ways 5 boys and 5 girls sit on a circle so that no two boys sit together?

If a polygon has 44 diagonals, then the number of its sides are:

The number of arrangement of r things out of n number of identical things is

The number of ways to fill the r^th position out of n distinct things in a row is