Page 173 - Fixtures
P. 173
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.
nn n −1
.C(2n,n − 1)
C(n,n − k)C(n,k − 1) k −1 n −k
n −1
.k = 1, ,n
. 2n
∑. n k 2C (n,k) = n(n + 1)2n−2 ()
k =1
nn
∑ k2C (n,k) = ∑ k2C (n,k)
k =1 k =0
nn
= ∑ (k2 − k)C (n,k) + ∑ kC (n,k)
k=0 k=0
∑ ∑= n−2 n(n − 1)(k2 − k)C (n − 2,k − 2) + n−1 n × kC(n − 1,k − 1)
k =2 k(k − 1) k =1 k
n−2 n−1
= n(n − 1)∑C (n − 2,k − 2) + n∑C(n − 1,k − 1)
k =2 k =1
= n(n − 1)2n−2 + n2n−1 = n(n + 1)2n−2
[AIME 1992] ( )
3:4:5
.n
