Permutations and Combinations

The number of distinct arrangements of all letters of the word 'EXAMINATION' is:

1. a\(\dfrac{11!}{3!}\)
2. b\(11!\)
3. c\(\dfrac{11!}{2! \cdot 2! \cdot 2!}\)
4. d\(\dfrac{11!}{2!}\)

How many 4-digit numbers can be formed using the digits 1 to 9 (no repetitions) that are divisible by 5?

1. a0
2. b336
3. c56
4. d3024

The number of ways in which 5 boys and 4 girls can be seated in a row such that all girls sit together is:

1. a\(5! \cdot 4!\)
2. b\(9!\)
3. c\(6! \cdot 4!\)
4. d\(5! + 4!\)

A polygon with \(n\) vertices has 54 diagonals. The number of sides is:

1. a10
2. b11
3. c13
4. d12

Assertion (A): The number of ways to choose a committee of 4 from 10 people is \(\binom{10}{4} = 210\).

Reason (R): By the formula \(\binom{n}{r} = n!/(r!\,(n - r)!)\), direct computation yields \(\binom{10}{4} = 210\).

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 ways are there to arrange the letters of the word 'PERMUTATIONS' so that all 7 consonants are together?

1. a\(8! \cdot 5!\)
2. b\(\dfrac{6! \cdot 7!}{2!}\)
3. c\(\dfrac{12!}{2!}\)
4. d\(6! \cdot 7!\)

How many committees of 5 can be formed from 6 men and 4 women if at least 2 women must be on the committee?

1. a56
2. b246
3. c120
4. d186

1. I. For all valid \(n, r\): \(\binom{n}{r} = \binom{n}{n - r}\).
2. II. For all valid \(n, r\): \(\binom{n}{r} + \binom{n}{r - 1} = \binom{n + 1}{r}\).
3. III. For all valid \(n, r\): \(P(n, r) = r! \cdot \binom{n}{r}\).

Consider the three identities listed above on factorials and combinations. Which are correct?

1. aAll three
2. bII and III
3. cOnly I
4. dI and II

The number of arrangements of the letters of 'MISSISSIPPI' is:

1. a\(11!\)
2. b\(\dfrac{11!}{4! \cdot 2!}\)
3. c\(\dfrac{11!}{4! \cdot 4! \cdot 2!}\)
4. d\(\dfrac{11!}{2! \cdot 4!}\)

In how many ways can 4 girls and 3 boys be seated in a row so that no two boys sit together?

1. a4320
2. b1440
3. c240
4. d144

Assertion (A): The number of 3-digit numbers that can be formed from the digits 1, 2, 3, 4 and 5, when repetition of the digits is allowed, is \({}^{5}P_{3} = 60\).

Reason (R): \({}^{n}P_{r}\) is the number of permutations of \(n\) different objects taken \(r\) at a time when repetition is not allowed.

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

If \(\binom{n}{r} = \binom{n}{r + 2}\), then \(r\) equals:

1. a\((n + 2)/2\)
2. b\((n - 2)/2\)
3. c\(n/2\)
4. d\(n - 2\)

Words formed using all letters of 'INDEPENDENCE' that begin with I and end with P number:

1. a\(\dfrac{10!}{3! \cdot 4! \cdot 2!}\)
2. b\(\dfrac{10!}{3! \cdot 4!}\)
3. c\(\dfrac{10!}{4! \cdot 2! \cdot 2!}\)
4. d\(10!\)

The total number of 4-letter codes from \(\{A, B, C, D, E\}\) with no repetition is:

1. a5
2. b60
3. c24
4. d120

How many 5-digit numbers greater than 50000 can be formed using digits \(1, 2, 3, 4, 5\) with no repetitions?

1. a72
2. b48
3. c24
4. d96

If \(P(n, 5) = 20 \cdot P(n, 3)\), then \(n\) equals:

1. a15
2. b10
3. c12
4. d8

Assertion (A): The number of straight lines determined by \(n\) points (no three collinear) is \(\binom{n}{2}\).

Reason (R): Two distinct points determine exactly one line, so the number of distinct lines equals the number of unordered pairs \(\binom{n}{2}\), provided no three points are collinear.

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

From a deck of 52 cards, the number of ways to choose 5 cards such that exactly 3 are aces is:

1. a\(\binom{4}{3} \cdot \binom{48}{2}\)
2. b\(\binom{4}{3} \cdot 48^{2}\)
3. c\(\binom{52}{5}\)
4. d\(P(4, 3) \cdot \binom{48}{2}\)

If \(\binom{n}{8} = \binom{n}{6}\), then \(\binom{n}{2}\) equals:

1. a105
2. b78
3. c91
4. d45

1. I. By convention, \(0! = 1\).
2. II. \(\binom{n}{0} = 1\) for any \(n \ge 0\).
3. III. \(P(n, n) = n!\).

Examine the three combinatorial conventions listed above. Which are correct?

1. aOnly I
2. bAll three
3. cI and II
4. dOnly II