Which relation lets 5! be rewritten as 5 times 4!?
An! equals n times (n - 1)!
Bn! equals (n - 1) times n!
Cn! equals n times (n + 1)!
Dn! equals (n + 1) times n!
Answer & Solution
Correct answer: A. n! equals n times (n - 1)!
1. A factorial can be defined recursively, meaning each value is built from the one below it.
2. The recursive step is n! = n times (n - 1)!, since the product from 1 to n is the product from 1 to n - 1 with one extra factor of n.
3. Applying it at n = 5 gives 5! = 5 times 4! = 5 times 24 = 120.
4. The options built on (n + 1)! move upward rather than downward, so they would need a value larger than the one being defined.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.1 Sequences and Their Notations_
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