Evaluate the following:

¹⁴C₃

Evaluate the following:

¹²C₁₀

Evaluate the following:

³⁵C₃₅

Evaluate the following:

Evaluate the following:

If ⁿC₁₂ = ⁿC₅, find the value of n.

If ⁿC₄ = ⁿC₆, find ¹²C_n.

If ⁿC₁₀ = ⁿC₁₂, find ²³C_n.

If ²⁴C_x = ²⁴C_{2x + 3}, find x.

If ¹⁸C_x = ¹⁸C_{x + 2}, find x.

If ¹⁵C_3r = ¹⁵C_{r + 3}, find r.

If ⁸C_r – ⁷C₃ = ⁷C₂, find r.

If ¹⁵C_r : ¹⁵C_{r – 1} = 11 : 5, find r.

If ^{n + 2}C₈ : ^{n – 2}P₄ = 57 : 16, find n.

If ²⁸C_2r : ²⁴C_{2r – 4} : 225 : 11, find r.

If ⁿC₄, ⁿC_5, and ⁿC₆ are in A.P., then find n.

If ²ⁿC₃ : ⁿC₂ = 44 : 3, find n.

If ¹⁶C_r = ¹⁶C_{r + 2}, find ^rC₄.

If α = ^mC₂, then find the value of ^αC₂.

Prove that the product of 2n consecutive negative integers is divisible by (2n)!

Prove that : ⁴ⁿC_2n : ²ⁿC_n = [1 . 3 . 5 …. (4n – 1)] : [1.3.5….. (2n – 1)]².

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 – 1}C_{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 + 2ⁿC_{r – 1} + ⁿC_{r – 2} = ^{n + 2}Cr

From a group of 15 cricket players, a team of 11 players is to be chosen. In how many ways can this be done?

How many different boat parties of 8, consisting of 5 boys and 3 girls, can be made from 25 boys and 10 girls?

In how many ways can a student choose 5 courses out of 9 courses if 2 courses are compulsory for every student?

In how many ways can a football team of 11 players be selected from 16 players? How many of these will (i) include 2 particular players? (ii) 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 products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?

From a class of 12 boys and 10 girls, 10 students are to be chosen for the competition, at least including 4 boys and 4 girls. The 2 girls who won the prizes last year should be included. In how many ways can the selection be made?

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 (ii) to include at least one officer?

A sports team of 11 students is to be constituted, choosing at least 5 from class XI and at least 5 from class XII. If there are 20 students in each of these classes, in how many ways can the teams be constituted?

A student has to answer 10 questions, choosing at least 4 from each of part A and part B. If there are 6 questions in part A and 7 in part B, in how many ways can the student choose 10 questions?

In an examination, a student to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Determine the number of ways in which the student can make a choice.

A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. In how many ways can he choose the 7 questions?

There are 10 points in a plane of which 4 are collinear. How many different straight lines can be drawn by joining these points.

Find the number of diagonals of

a hexagon

Find the number of diagonals of

a polygon of 16 sides

How many triangles can be obtained by joining 12 points, five of which are collinear?

In how many ways can a committee of 5 persons be formed out of 6 men and 4 women when at least one woman has to be necessarily selected?

In a village, there are 87 families of which 52 families have at most 2 children. In a rural development programme, 20 families are to be helped chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made?

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? (ii) at least one boy and one girl? (ii) at least 3 girls?

A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of this committee would consist of 1 man and 2 women?

Find the number of (i) diagonals (ii) triangles formed in a decagon.

Determine the number of 5 cards combinations out of a deck of 52 cards if at least one of the 5 cards has to be a king?

We wish to select 6 persons from 8, but if the person A is chosen, then B must be chosen. In how many ways can the selection be made?

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls, and 5 blue balls if each selection consists of 3 balls of each color.

Determine the number of 5 cards combinations out of a deck of 52 cards if there is exactly one ace in each combination.

In how many ways can one select a cricket team of eleven from 17 players in which only 5 persons can bowl if each cricket team of 11 must include exactly 4 bowlers?

A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

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 :

at most 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 help a student select the questions?

A parallelogram is cut by two sets of m lines parallel to its sides. Find the number of parallelograms thus formed.

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 (ii) triangles can be formed by joining them?

How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?

There are 10 persons named P₁, P₂, P₃,……P₁₀. Out of 10 persons, 5 persons are to be arranged in a line such that is 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

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 the first letter is a vowel?

Find the number of permutations of n distinct things taken r together, in which 3 particular things must occur together.

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?

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

Find the number of ways in which: (a) a selection (b) an arrangement, of four letters, can be made from the letters of the word ‘PROPORTION’?

How many words can be formed by taking 4 letters at a time from the letters of the word ‘MORADABAD’?

A businessman hosts a dinner to 21 guests. He has 2 round tables which can accommodate 15 and 6 persons each. In how many ways can he arrange the guests?

Find the number of combinations and permutations of 4 letters taken from the word ‘EXAMINATION.’

A tea party is arranged for 16 persons along two sides of a long table with 8 chairs on each side. Four persons wish to sit on one particular side and two on the other side. In how many ways can they are seated?

If , then write the values of n.

Write the number of diagonals of an n-sided polygon.

Write the expression in the simplest form.

Write the value of .

There are 3 letters and 3 directed envelopes. Write the number of ways in which no letter is put in the correct envelope.

Write the maximum number of points of intersection of 8 straight lines in a plane.

Write the number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.

Write the number of ways in which 12 boys may be divide into three groups of 4 boys each.

Write the total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants.

Mark the correct alternative in the following:

If , then is equal to

Mark the correct alternative in the following:

If , then is equal to

Mark the correct alternative in the following:

If , then r is equal to

Mark the correct alternative in the following:

If , then r is equal to

Mark the correct alternative in the following:

If , then is equal to

Mark the correct alternative in the following:

If, then

Mark the correct alternative in the following:

If , then n =

Mark the correct alternative in the following:

If , then x =

Mark the correct alternative in the following:

If , then a =

Mark the correct alternative in the following:

is equal to

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

If , then is equal to

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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 point is

Mark the correct alternative in the following:

How many different committees of 5 can be formed from 6 men and 4 women on which exact 3 men and 2 women serve?

Mark the correct alternative in the following:

If , then the value of r is

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

The value of is

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

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

Mark the correct alternative in the following:

If , then n =

Mark the correct alternative in the following:

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