Chapter 11 - 11.6 - Counting Principles - 11.6 Exercises - Page 825: 79

$n=6$ $n=5$

$14·~_nP_3=~_{n+2}P_4$ $14·\frac{n!}{(n-3)!}=\frac{(n+2)!}{(n+2-4)!}$ $14·\frac{n(n-1)(n-2)(n-3)!}{(n-3)!}=\frac{(n+2)(n+1)n(n-1)(n-2)!}{(n-2)!}$ $14·n(n-1)(n-2)=(n+2)(n+1)n(n-1)$ $14(n-2)=(n+2)(n+1)$ $14n-28=n^2+3n+2$ $0=n^2-11n+30$ $0=n^2-5n-6n+30$ $0=n(n-5)-6(n-5)$ $0=(n-6)(n-5)$ $n-6=0$ $n=6$ $n-5=0$ $n=5$