RD Sharma solutions for Mathematics [English] Class 11 chapter 17 - Combinations [Latest edition]

Evaluate the following:

¹⁴C₃

Evaluate the following:

¹²C₁₀

Evaluate the following:

³⁵C₃₅

Evaluate the following:

^n + 1C_n

Evaluate the following:

For all positive integers n, show that ²ⁿC_n + ²ⁿC_{n − 1} = `1/2` ^{2n + 2}C_n+1 

Evaluate

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:
n · _^n − 1C_{r − 1} = (n − r + 1) ⁿC_{r − 1}

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

Let r and n be positive integers such that 1 ≤ r ≤ n. Then prove the following:

 ⁿC_r + 2 · ⁿC_{r − 1} + ⁿC_{r − 2} = ^n + 2C_r.

In how many ways can a football team of 11 players be selected from 16 players? How many of these will

include 2 particular players?

In how many ways can a football team of 11 players be selected from 16 players? How many of these will

 exclude 2 particular players?

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:
a particular professor is included.

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

a particular student is included.

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees:

a particular student is excluded.

How many different selections of 4 books can be made from 10 different books, if
there is no restriction;

How many different selections of 4 books can be made from 10 different books, if
two particular books are always selected;

How many different selections of 4 books can be made from 10 different books, if two particular books are never selected?

From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer

From 4 officers and 8 jawans in how many ways can 6 be chosen. to include at least one officer?

Find the number of diagonals of , 1.a hexagon

Find the number of diagonals of (ii) a polygon of 16 sides.

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl?

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (ii) at least one boy and one girl? 

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has(iii) at least 3 girls? 

Find the number of (i) diagonals

Find the number of (ii) triangles

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: exactly 3 girls?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: at least 3 girls?

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: atmost 3 girls?

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (i) straight lines

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many (ii) triangles can be formed by joining them?

There are 10 persons named\[P_1 , P_2 , P_3 , . . . . , P_{10}\]
Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P₁ must occur whereas P₄ and P₅ do not occur. Find the number of such possible arrangements.

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if (i) 4 letters are used at a time 

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if  all letters are used at a time 

How many words, with or without meaning can be formed from the letters of the word 'MONDAY', assuming that no letter is repeated, if all letters are used but first letter is a vowel?

Find the number of permutations of n different things taken r at a time such that two specified things occur together?

Find the number of ways in which : (a) a selection

Find the number of ways in which : (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

Write \[\sum^m_{r = 0} \ ^{n + r}{}{C}_r\] in the simplified form.

Write the value of\[\sum^6_{r = 1} \ ^{56 - r}{}{C}_3 + \ ^ {50}{}{C}_4\]

If ²⁰C_r = ²⁰C_r−10, then ¹⁸C_r is equal to

If ²⁰C_r = ²⁰C_{r + 4} , then ^rC₃ is equal to

If ¹⁵C_3r = ¹⁵C_{r + 3} , then r is equal to

If ²⁰C_{r + 1} = ²⁰C_{r − 1} , then r is equal to

If C (n, 12) = C (n, 8), then C (22, n) is equal to

If ^mC₁ = ⁿC₂ , then

If ⁿC₁₂ = ⁿC₈ , then n =

If ⁿC_r + ⁿC_{r + 1} = ^n + 1C_x , then x =

If\[\ ^{( a^2 - a)}{}{C}_2 = \ ^{( a^2 - a)}{}{C}_4\] , then a =

⁵C₁ + ⁵C₂ + ⁵C₃ + ⁵C₄ +⁵C₅ is equal to

Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to

There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of them are collinear, is

Three persons enter a railway compartment. If there are 5 seats vacant, in how many ways can they take these seats?

In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women?

There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two of them is

There are 13 players of cricket, out of which 4 are bowlers. In how many ways a team of eleven be selected from them so as to include at least two bowlers?

If C₀ + C₁ + C₂ + ... + C_n = 256, then ²ⁿC₂ is equal to

The number of ways in which a host lady can invite for a party of 8 out of 12 people of whom two do not want to attend the party together is

Given 11 points, of which 5 lie on one circle, other than these 5, no 4 lie on one circle. Then the number of circles that can be drawn so that each contains at least 3 of the given points is

How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?
(a) 6
(b) 20
(c) 60
(d) 120

If ⁴³C_{r − 6} = ⁴³C_{3r + 1} , then the value of r is

The number of diagonals that can be drawn by joining the vertices of an octagon is

The value of\[\left( \ ^{7}{}{C}_0 + \ ^{7}{}{C}_1 \right) + \left( \ ^{7}{}{C}_1 + \ ^{7}{}{C}_2 \right) + . . . + \left( \ ^{7}{}{C}_6 + \ ^{7}{}{C}_7 \right)\] is

Among 14 players, 5 are bowlers. In how many ways a team of 11 may be formed with at least 4 bowlers?

A lady gives a dinner party for six guests. The number of ways in which they may be selected from among ten friends if two of the friends will not attend the party together is

If ^n + 1C₃ = 2 · ⁿC₂ , then n =

The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is