F.5 Permutation&Combination4

Thecoach of a school athletic team is going to select 4 players for a 4x100m relayrace
from n members of the team．It isgiven that Franky must be selected，but the leg of the
race he is responsible for has not yet been determined．If the coach has 2016 ways to
arrange the legs of the race，find the value of n.
Sol
4－1=3
4!=24
2016/24=84
C(n，3)=n!/[n－3)!3!]==n(n－1)(n－2)/6=84
n(n－1)(n－2)=6*84=504
8*7*6=336
9*8*7=504
So
n=9

(n-1)C3 = 84
(n - 1)(n - 2)(n - 3) = 504
n^3 - 6n^2 + 11n - 510 = 0
(n - 10)(n^2 + 4n + 51) = 0
n = 10 or no solution