ASVAB Study GuideDivisibility Rules for 3 and 9
Definition & Calculation
$$n! = n \times (n - 1) \times \ldots \times 2 \times 1$$
(for n ∈ ℕ*)
A factorial is the product of all positive integers less than or equal to n.
Example
Calculate $$5!$$ and simplify $$\frac{7!}{5!}$$
Show solution
1) $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$.
2) $$\frac{7!}{5!} = \frac{(7 \times 6 \times 5!)}{5!} = 7 \times 6 = 42$$
Practice problems
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Problem 1
What is the value of $$4!$$?
Show solution
Answer: 24
$$4! = 4 \times 3 \times 2 \times 1 = 24$$
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Problem 2
Simplify the expression: $$\frac{6!}{(4! \times 2!)}$$
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Answer: 15
$$\frac{6!}{(4! \times 2!)} = \frac{(6 \times 5 \times 4!)}{(4! \times 2)} = \frac{(6 \times 5)}{2} = 15$$
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Problem 3
Find the value of n if $$\frac{(n + 1)!}{n!} = 8$$
Show solution
Answer: 7
$$\frac{(n + 1)!}{n!} = (n + 1) \times n! / n! = n + 1$$. Therefore, $$n + 1 = 8 \Rightarrow n = 7$$
Special cases:
- $$0! = 1$$ (by definition)
- $$1! = 1$$
Factorials grow extremely fast!