EQUATION:
P(16,4) = 208 · P(n,2)
FORMULA:
[tex]\text{P}(n,r) = \frac{n!}{(n-r)!}[/tex]
SOLUTION:
[tex]\text{P}(16,4) = 208 \times \text{P}(n,2)[/tex]
[tex]\frac{16!}{(16-4)!} = 208 \times \frac{n!}{(n-2)!}[/tex]
[tex]\frac{16!}{12!} = 208 \times \frac{n(n-1)\cancel{(n-2)!}}{\cancel{(n-2)!}}[/tex]
[tex]13 \times 14 \times 15 \times 16 = 208 \times n(n-1)[/tex]
[tex]\cancel{208} \times 14 \times 15 = \cancel{208} \times (n^2 - n)[/tex]
[tex]14 \times 15 = n^2-n[/tex]
[tex]210 = n^2 - n[/tex]
[tex]n^2 - n - 210 = 0[/tex]
[tex](n-15)(n+14) = 0[/tex]
[tex]\therefore \begin{cases} n - 15 = 0 \implies n = 15\\ n + 14 = 0 \implies n = -14\\ \end{cases}[/tex]
[tex]\textsf{Note that} \: n \in \mathbb{N}[/tex]
[tex]\textsf{So }n \neq -14.[/tex]
[tex]\textsf{Hence, n = 15}[/tex]
ANSWER:
n = 15
