Combinatorics: Permutations and Combinations

The value of \(\dfrac{8!}{6!}\) is:

1. a\(\dfrac{4}{3}\)
2. b2
3. c48
4. d56

Meera has 3 shirts and 4 pairs of trousers. In how many different ways can she choose one shirt and one pair of trousers to wear?

1. a12
2. b7
3. c24
4. d64

The value of \(^{8}P_{3}\) is:

1. a56
2. b24
3. c336
4. d512

1. I. \(^{n}C_{0} = 1\)
2. II. \(^{n}C_{n} = n\)
3. III. \(^{n}C_{1} = n\)

For a natural number \(n\), which of the statements is/are correct?

1. aOnly I
2. bOnly I and III
3. cOnly II and III
4. dI, II and III

Assertion (A): \(^{12}C_{9} = {}^{12}C_{3}\)

Reason (R): Choosing \(r\) objects out of \(n\) is the same as choosing the \(n - r\) objects to be left out, so \(^{n}C_{r} = {}^{n}C_{n-r}\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

How many 3-digit numbers can be formed using the digits 1, 2, 3, 4 and 5 if no digit is repeated?

1. a125
2. b10
3. c15
4. d60

In how many ways can the letters of the word MONDAY be arranged so that each arrangement begins with M and ends with Y?

1. a720
2. b24
3. c120
4. d48

1. I. \(0! = 1\)
2. II. \((2 + 3)! = 2! + 3!\)
3. III. \(n! = n \times (n - 1)!\) for every natural number \(n\)

About factorials, which of the statements is/are correct?

1. aOnly I
2. bOnly II
3. cOnly I and III
4. dI, II and III

A committee of 3 men and 2 women is to be formed from 5 men and 4 women. In how many ways can this be done?

1. a60
2. b16
3. c126
4. d720

Match each expression in List I with its value in List II:

1. a(i)-S, (ii)-R, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-S, (iii)-Q, (iv)-P
3. c(i)-R, (ii)-P, (iii)-S, (iv)-Q
4. d(i)-R, (ii)-S, (iii)-P, (iv)-Q

If \(^{n}C_{2} = 28\), then \(n\) equals:

1. a7
2. b14
3. c8
4. d56

Assertion (A): The number of ways of selecting 3 students out of 7 for a quiz team is 35.

Reason (R): In a selection, the order in which the objects are chosen matters.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

At a meeting of 12 people, each person shakes hands exactly once with every other person. How many handshakes take place?

1. a132
2. b66
3. c144
4. d11

1. I. Choosing 4 members for a committee from 10 people is a combination problem.
2. II. Arranging 4 people in the front row of a photograph is a permutation problem.
3. III. Choosing a president and a secretary from a club of 20 members is a combination problem.

Which of these statements about permutations and combinations is/are correct?

1. aOnly I and II
2. bOnly I
3. cOnly II and III
4. dI, II and III

How many diagonals does a polygon with 10 sides have?

1. a45
2. b90
3. c70
4. d35

Match each situation in List I with its number of ways in List II:

1. a(i)-S, (ii)-Q, (iii)-R, (iv)-P
2. b(i)-R, (ii)-Q, (iii)-S, (iv)-P
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-R, (ii)-Q, (iii)-P, (iv)-S

Arjun's bicycle lock has 4 rings, each showing the digits 0 to 9. The code is one digit on each ring, and digits may repeat. How many different codes are possible?

1. a5040
2. b40
3. c10000
4. d210

Assertion (A): The number of ways in which 4 people can stand in a queue is \(4^{4} = 256\).

Reason (R): Once a person takes a place in the queue, one person fewer is left to choose from for the next place.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

A class of 30 students must elect a monitor and a deputy monitor, and one student cannot hold both posts. In how many ways can the two posts be filled?

1. a870
2. b435
3. c900
4. d59

In how many ways can the letters of the word DELHI be arranged so that the two vowels are always together?

1. a24
2. b48
3. c120
4. d72