Page 177 - Fixtures

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n!

C (n, k ) = C(n,k) = k !(n − k)!

C (n,k) + C (n,k + 1) C (n + 1,k + 1) (n + 1)!

(k + 1)!(n − k)!

= n !(k + 1)! = k + 1
k !(n + 1)! n + 1

n−1  C (n,k)  3 = n −1  k + 1 3 = (n 1 n −1 + 1)3
+ 1)  k=0 n + 1 + 1)3
∑ ∑ ∑k=0 C(n,k) + C (n,k (k

k=0

= 1 [13 + 23 + + n3 ] = 1 × n2(n + 1)2 = n2
(n + 1)3 (n + 1)3 4 4(n + 1)

,

n2 = 4 n
4(n + 1) 5
5n2 − 16n − 16 = 0

(5n + 4)(n − 4) = 0

.n = 4