Exercise · ZW-FACTORIAL-004 · ★☆☆ Basic
Why is \(0!\) defined as \(1\)?
Choose one option.
Show answer
- B. Because then \(n!=n\,(n-1)!\) also holds for \(n=1\), and there is exactly one way to arrange zero objects.
Explanation
With \(0!=1\) we get \(1!=1\cdot 0!\), and the count of arrangements of no objects is 1 (the empty list). It is neither zero nor a limit.
Review the concept: Factorial
The factorial of an integer \(n\ge 0\), written \(n!\), is the product of the positive integers less than or equal to \(n\); that of zero is one by convention.
\[n!=n(n-1)\cdots 2\cdot 1,\quad 0!=1\]
More practice problems: factorial
- ZW-FACTORIAL-001 Compute 5!.
- ZW-FACTORIAL-002 In how many ways can 6 different people line up in a row?
- ZW-FACTORIAL-003 Compute (6!)/(4!).
- ZW-FACTORIAL-005 For n≥ 0, which equals (n+1)!?
- ZW-FACTORIAL-006 Compute (8!)/(6! 2!).
- ZW-FACTORIAL-007 Simplify (n!)/((n-2)!) for n≥ 2.
- ZW-FACTORIAL-008 Words of 4 letters are formed from A, B, C, D (no repetition). How many start with A?
- ZW-FACTORIAL-009 Five people (3 men and 2 women) line up with the two women next to each other. In how many ways?