## MCQ (Single Correct Answer)

Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls...

In a high school, a committee has to be formed from a group of 6 boys M1, M2, M3, M4, M5, M6 and 5 girls G1, G2, G3, G4, G5.(i) Let $$\alpha $$1 be th...

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be select from this club including the selection of a captain (from among thes...

Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and ...

Let $${{a_n}}$$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Le...

Let $${{a_n}}$$ denote the number of all n-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them are 0.Le...

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements/Expressions in Column I with the Statements/Expressions ...

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of ...

A rectangle with sides of lenght (2m - 1) and (2n - 1) units is divided into squares of unit lenght by drawing parallel lines as shown in the diagram,...

If the LCM of p, q is $${r^2}\,{r^4}\,{s^2}$$, where r, s, t are prime numbers and p, q are the positive integers then number of ordered pair (p, q) i...

The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is

Let $${T_n}$$ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If $${T_{n + 1}} - {T_n} = 21$$, ...

How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?

A five-digit numbers divisible by 3 is to be formed using the numerals 0, 1, 2, 3, 4 and 5, without repetition. The total number of ways this can be d...

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the numbers of words which have at l...

Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs mar...

The value of the expression $$\,{}^{47}{C_4} + \sum\limits_{j = 1}^5 {^{52 - j}\,{C_3}} $$ is equal to

$${}^n{C_{r - 1}} = 36,{}^n{C_r} = 84\,\,and\,\,{}^n{C_{r + 1}} = 126$$, then r is :

## Numerical

The number of 4-digit integers in the closed interval [2022, 4482] formed by using the digits $$0,2,3,4,6,7$$ is _________.

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take p...

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least ...

Let |X| denote the number of elements in a set X. Let S = {1, 2, 3, 4, 5, 6} be a sample space, where each element is equally likely to occur. If A an...

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, th...

The number of 5 digit numbers which are divisible by 4, with digits from the set {1, 2, 3, 4, 5} and the repetition of digits is allowed, is ............

Words of length 10 are formed using the letters A, B, C, D, E, F, G, H, I, J. Let x be the number of such words where no letter is repeated; and let y...

Let n be the number of ways in which 5 boys and 5 girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let m ...

Let $${n_1}\, < {n_2}\, < \,{n_3}\, < \,{n_4}\, < {n_5}$$ be positive integers such that $${n_1}\, + {n_2}\, + \,{n_3}\, + \,{n_4}\, + {n_...

Let $${n \ge 2}$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining ev...

Consider the set of eight vectors $$V = \left\{ {a\,\hat i + b\,\hat j + c\hat k:a,\,b,\,c\, \in \left\{ { - 1,\,1} \right\}} \right\}$$. Three non-co...

Let $$\left( {x,\,y,\,z} \right)$$ be points with integer coordinates satisfying the system of homogeneous equation:
$$$\matrix{
{3x - y - z = 0} ...

## MCQ (More than One Correct Answer)

Let $${S_1} = \left\{ {(i,j,k):i,j,k \in \{ 1,2,....,10\} } \right\}$$,$${S_2} = \left\{ {(i,j):1 \le i < j + 2 \le 10,i,j \in \{ 1,2,...,10\} } \r...

An n-digit number is a positive number with exactly digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 an...

## Subjective

If total number of runs scored in n matches is $$\left( {{{n + 1} \over 4}} \right)\,\,({2^{n + 1}} - n - 2)\,$$ where $$n > 1$$, and the runs scor...

Prove by permulation or otherwise $${{({n^2})!} \over {{{(n!)}^n}}}$$ is an integer $$(n \in {1^ + })$$.

A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to included in a committee? In...

Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on ...

A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black bal...

7 relatives of a man comprises 4 ladies and 3 gentlemen ; his wife has also 7 relatives ; 3 of them are ladies and 4 gentlemen. In how many ways can t...

m men and n women are to be seated in a row so that no two women sit together. If $$m > n$$, then show that the number of ways in which they can be...

Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can be place...

Six X' s have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be ...

## Fill in the Blanks

Total number of ways in which six ' + ' and four ' - ' signs can be arranged in a line such that no two ' - ' signs occur together is....................

There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a ...

The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using t...

In a certain test, $${a_i}$$ students gave wrong answers to atleast i questions, where i = 1, 2,..., k. No student gave more than k wrong answers. Th...

## True or False

The product of any r consecutive natural numbers is always divisible by r!