Permutations and Combinations Questions

MathematicsPermutations and Combinations57 Questions Found

MathematicsFundamental Principle of CountingJEE Advanced 2012Moderate

Comprehension Passage

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. Let

b_{n}

= the number of such

n

-digit integers ending with digit 1 and

c_{n}

= the number of such

n

-digit integers ending with digit 0.

Question 1:

The value of $b_{6}$ is

(A)

(B)

(C)

(D)

Answer:B

Question 2:

Which of the following is correct?

(A)

a_{17} = a_{16} + a_{15}

(B)

c_{17} \neq = c_{16} + c_{15}

(C)

b_{17} \neq = b_{16} + c_{16}

(D)

a_{17} = c_{17} + b_{16}

Answer:A

MathematicsLinear PermutationsJEE Advanced 2008Moderate

View in:English Hindi

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS.

List-I

(P)

The number of permutations containing the word ENDEA is

(Q)

The number of permutations in which the letter E occurs in the first and the last positions is

(R)

The number of permutations in which none of the letters D, L, N occurs in the last five positions is

(S)

The number of permutations in which the letters A, E, O occur only in odd positions is

List-II

(1)

5!

(2)

2 \times 5!

(3)

7 \times 5!

(4)

21 \times 5!

Answer:P → 1, Q → 4, R → 2, S → 2

MathematicsLinear PermutationsJEE Advanced 1983Easy

View in:English Hindi

$m$ 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 seated is $\frac{m ! ( m + 1 )!}{( m - n + 1 )!}$

MathematicsCombinations and SelectionJEE Advanced 1994Easy

View in:English Hindi

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?

MathematicsLinear PermutationsJEE Advanced 2004Moderate

View in:English Hindi

Prove by permutation or otherwise $\frac{( n ^{2} )!}{( n ! ) ^{n}}$ is an integer ( $n \in I^{+}$ ).

MathematicsFundamental Principle of CountingJEE Advanced 1982Moderate

View in:English Hindi

In a certain test, $a_{i}$ students gave wrong answers to atleast $i$ questions, where $i = 1, 2, \dots, k$ . No student gave more than $k$ wrong answers. The total number of wrong answers given is .........

Answer:

a_{1} + a_{2} + \dots + a_{k}

MathematicsCombinations and SelectionJEE Advanced 1991Easy

View in:English Hindi

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.

Answer:

^{11} C_{5} \times 9! \times 9!

MathematicsCombinations and SelectionJEE Advanced 1996Easy

View in:English Hindi

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 .........

Answer:

(k - 1 n - \frac{k ( k + 1 )}{2} + k - 1)

MathematicsDerangement PrincipleJEE AdvancedEasy

View in:English Hindi

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 box, could be placed such that a ball does not go to a box of its own colour is .........

Answer:9

MathematicsCombinations and SelectionJEE Advanced 1978Easy

View in:English Hindi

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.

Answer:26

...6

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of b6​ is

Which of the following is correct?

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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 seated is (m−n+1)!m!(m+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 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; } Prove by permutation or otherwise (n!)n(n2)!​ is an integer (n∈I+).

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.

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 box, could be placed such that a ball does not go to a box of its own colour is .........

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.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of b6​ is

Which of the following is correct?

List-I

List-II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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 seated is (m−n+1)!m!(m+1)!​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Prove by permutation or otherwise (n!)n(n2)!​ is an integer (n∈I+).

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.

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 box, could be placed such that a ball does not go to a box of its own colour is .........

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.

The value of $b_{6}$ is

$m$ 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 seated is $\frac{m ! ( m + 1 )!}{( m - n + 1 )!}$

Prove by permutation or otherwise $\frac{( n ^{2} )!}{( n ! ) ^{n}}$ is an integer ( $n \in I^{+}$ ).

The value of $b_{6}$ is

$m$ 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 seated is $\frac{m ! ( m + 1 )!}{( m - n + 1 )!}$

Prove by permutation or otherwise $\frac{( n ^{2} )!}{( n ! ) ^{n}}$ is an integer ( $n \in I^{+}$ ).