Combinations and Selection

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JEE Main 2013

LEVELBoard

Let $T_{n}$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n + 1} - T_{n} = 10$ , then the value of $n$ is :

These are 10 points in a plane, out of these 6 are collinear, if $N$ is the number of triangles formed by joining these points. then:

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is :

Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is $^{9} C_{3}$ . Statement-2: The number of ways of choosing any 3 places from 9 different places is $^{9} C_{3}$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2010

LEVELBoard

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement-1 : The number of different ways the child can buy the six ice-creams is $^{10} C_{5}$ . Statement-2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 $A$ 's and 4 $B$ 's in a row.

(A)

Statement-1 is false, Statement-2 is true

(B)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1

(C)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1

(D)

Statement-1 is true, Statement-2 is false

JEE Main 2006

LEVELBoard

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

If $^{n} C_{r}$ denotes the number of combination of $n$ things taken $r$ at a time, then the expression $^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$ equals

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

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

Let $S = {1, 2, 3, 4}$ . The total number of unordered pairs of disjoint subsets of $S$ is equal to

If $^{n - 1} C_{r} = (k^{2} - 3)^{n} C_{r + 1}$ , then $k \in$

(A)

(- \infty, - 2] \cup [2, \infty)

(B)

[2, \infty)

(C)

[- 3, 3]

(D)

(3, 2]

JEE Advanced 2001

LEVELJEE Main

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

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

Let $n_{1} < n_{2} < n_{3} < n_{4} < n_{5}$ be positive integers such that $n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 20$ . Then the number of such distinct arrangements $(n_{1}, n_{2}, n_{3}, n_{4}, n_{5})$ is

JEE Advanced 1986

LEVELJEE Main

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 ball is to be included in the draw?

JEE Advanced 1985

LEVELJEE Main

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 they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relatives and 3 of the wife's relatives?

JEE Advanced 1984

LEVELJEE Main

The side $A B, B C$ and $C A$ of a triangle $A B C$ have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices is .........

JEE Advanced 1981

LEVELJEE Main

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 we place the balls so that no box remains empty ?

JEE Advanced 1978

LEVELJEE Main

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 done.

JEE Advanced 1996

LEVELJEE Main

Let $n$ and $k$ be positive such that $n \geq \frac{k ( k + 1 )}{2}$ . The number of solutions $(x_{1}, x_{2}, \dots, x_{k})$ , $x_{1} \geq 1, x_{2} \geq 2, \dots, x_{k} \geq k$ , all integers, satisfying $x_{1} + x_{2} + \dots + x_{k} = n$ , is .........

JEE Advanced 1991

LEVELJEE Main

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 the other side. Determine the number of ways in which the sitting arrangements can be made.

JEE Advanced 1994

LEVELJEE Main

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 be included in a committee? In how many of these committees (a) The women are in majority? (b) The men are in majority?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tn​ be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1​−Tn​=10, then the value of n is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } These are 10 points in a plane, out of these 6 are collinear, if N is the number of triangles formed by joining these points. then:

Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is :

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If nCr​ denotes the number of combination of n things taken r at a time, then the expression nCr+1​+nCr−1​+2×nCr​ equals

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={1,2,3,4}. The total number of unordered pairs of disjoint subsets of S is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If n−1Cr​=(k2−3)nCr+1​, then k∈

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Tn​ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn+1​−Tn​=21, then n equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } nCr−1​=36,nCr​=84 and nCr+1​=126, then r is:

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 ball is to be included in the draw?

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 they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relatives and 3 of the wife's relatives?

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 we place the balls so that no box remains empty ?

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 done.

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 the other side. Determine the number of ways in which the sitting arrangements can be made.

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 be included in a committee? In how many of these committees (a) The women are in majority? (b) The men are in majority?

Let $T_{n}$ be the number of all possible triangles formed by joining vertices of an $n$ -sided regular polygon. If $T_{n + 1} - T_{n} = 10$ , then the value of $n$ is :

These are 10 points in a plane, out of these 6 are collinear, if $N$ is the number of triangles formed by joining these points. then:

If $^{n} C_{r}$ denotes the number of combination of $n$ things taken $r$ at a time, then the expression $^{n} C_{r + 1} +^{n} C_{r - 1} + 2 \times^{n} C_{r}$ equals

Let $S = {1, 2, 3, 4}$ . The total number of unordered pairs of disjoint subsets of $S$ is equal to

If $^{n - 1} C_{r} = (k^{2} - 3)^{n} C_{r + 1}$ , then $k \in$

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

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