Question 1

What is the value of $5!$ (5 factorial)?

Accepted Answer

The factorial of a positive integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. Therefore, $5! = 5 	imes 4 	imes 3 	imes 2 	imes 1 = 120$. Thus, option B is correct.

Question 2

Evaluate $^5P2$.

Accepted Answer

The formula for permutation is $^nPr = \frac{n!}{(n-r)!}$. Substituting $n = 5$ and $r = 2$, we get $^5P2 = \frac{5!}{(5-2)!} = \frac{5!}{3!} = \frac{5 	imes 4 	imes 3!}{3!} = 5 	imes 4 = 20$. Therefore, option A is correct.

Question 3

Evaluate $^6C4$.

Accepted Answer

The formula for combination is $^nCr = \frac{n!}{r!(n-r)!}$. Substituting $n = 6$ and $r = 4$, we get $^6C4 = \frac{6!}{4!(6-4)!} = \frac{6!}{4!2!} = \frac{6 	imes 5 	imes 4!}{4! 	imes 2 	imes 1} = \frac{30}{2} = 15$. Therefore, option C is correct.

Question 4

If $^nC8 = ^nC2$, find the value of $n$.

Accepted Answer

We know the property of combinations: if $^nCa = ^nCb$, then either $a = b$ or $n = a + b$. Here, $8 
eq 2$, so we must have $n = 8 + 2 = 10$. Therefore, option B is correct.

Question 5

What is the value of $0!$?

Accepted Answer

$0!$ (zero factorial) is defined as 1 by convention, which is consistent with the formulas of permutations and combinations. Thus, option A is correct.

Permutations and Combinations