Page 172 - Fixtures

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S = C(n,1) + 3C(n, 3) + 5C(n,5) + + (n − 1)C(n,n − 1)
C(n,k) = C(n,n − k)

S = (n − 1)C(n,1) + (n − 3)C(n, 3) + + 1C(n,1)

2S = n[C(n,1) + C(n, 3) + + C(n,n − 1)]

C(n, 0) +C(n,1) +C(n,2) +C(n, 3) + + C(n,n) = 2n

C(n,1) + C(n, 3) + + C(n,n − 1) = 2n = 2n−1
2

:S = n2n−2 . 2S = n2n−1 ,

:

2C(n,2) + 4C(n, 4) + 6(n,6) + + nC(n,n) = n2n−2

.n

. n − 1) = C (2n, n − 1) ()

∑C (n,k)C(n,k
k =1

C(n,k) = C(n,n − k)

n

∑C (n,n − k)C (n,k − 1) = C (2n,n − 1)

k =1