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Factorials

Source: 2020 AMC 12B #6

February 6, 2020
factorial2020 AMC 12BAMCAMC 12AMC 12 B2020 AMC

Problem Statement

For all integers n≥9,n \geq 9, the value of (n+2)!−(n+1)!n!\frac{(n+2)!-(n+1)!}{n!} is always which of the following?
<spanclass=′latex−bold′>(A)</span>a multiple of 4<spanclass=′latex−bold′>(B)</span>a multiple of 10<spanclass=′latex−bold′>(C)</span>a prime number<spanclass=′latex−bold′>(D)</span>a perfect square<spanclass=′latex−bold′>(E)</span>a perfect cube<span class='latex-bold'>(A) </span> \text{a multiple of }4 \qquad <span class='latex-bold'>(B) </span> \text{a multiple of }10 \qquad <span class='latex-bold'>(C) </span> \text{a prime number} \\ <span class='latex-bold'>(D) </span> \text{a perfect square} \qquad <span class='latex-bold'>(E) </span> \text{a perfect cube}
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