f4 Binomial Expansion

(n - 4)! = (n -4)(n - 5)(n -6)!
(n - 5)! = (n - 5)(n - 6)!
so n!/4!(n -4)! = [1/4!(n -4)(n-5)][n!/(n -6)!]
n!/6!(n-6)! = [1/6!][n!/(n -6)!]
n!/5!(n -5)! = [1/5!(n-5)][n!/(n -6)!]
Eliminating the common factor n!/(n -6)! we get
1/4!(n - 4)(n - 5) + 1/6! = 2/5!(n - 5)
1/(n - 4)(n - 5) + 1/30 = 2/5(n - 5) [ By eliminating the common factor 4!]
30 + ( n - 4)( n - 5) = 12(n - 4)
30 + n^2 - 9n + 20 = 12n - 48
n^2 - 21n + 98 = 0
(n - 7)(n - 14) = 0
so n = 7 or 14.

nC4+nC6=(nC5)(2)
n!/[(4!)(n-4)!] + n!/[(6!)(n-6)!] = 2(n!)/[(5!)(n-5)!]
[(n-4)!](n-3)(n-2)(n-1)n/[(4!)(n-4)!] + [(n-6)!](n-5)(n-4)(n-3)(n-2)(n-1)n/[(6!)(n-6)!] = 2[(n-5)!](n-4)(n-3)(n-2)(n-1)n/[(5!)(n-5)!]
n(n-1)(n-2)(n-3)/(4!) + n(n-1)(n-2)(n-3)(n-4)(n-5)/(6!) = 2n(n-1)(n-2)(n-3)(n-4)/(5!)
[n(n-1)(n-2)(n-3)/(4!) + n(n-1)(n-2)(n-3)(n-4)(n-5)/(6!)] x (6!)/n(n-1)(n-2)(n-3) = 2n(n-1)(n-2)(n-3)(n-4)/(5!) x (6!)/n(n-1)(n-2)(n-3)
5x6 + (n-4)(n-5) = 2(n-4) x 6
30 + n^2 - 5n -4n +20 = 12n -48
n^2 -21n + 98 = 0
n^2 - (7+14)n + (7)x(14) = 0
(n-7)(n-14)=0
n=7 or 14