Latest TCS Aptitude Question SOLUTION: What is∑ 28 k=0 k^2(28Ck), where 28Ck is the number of ways of choosing k items from 28 items?

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What is∑ 28 k=0 k^2(28Ck), where 28Ck is the number of ways of choosing k items from 28 items?

Asked In TCS vasu devu (7 years ago)

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jayavelu  (7 years ago)

∑ k^2 C(n,k) = ∑ {k(k-1)+k} C(n,k)
= ∑ k(k-1) C(n,k) + ∑ k C(n,k)
= ∑ k(k-1) {n(n-1) /k(k-1)} C(n-2,k-2) + ∑ k {n/k) C(n-1,k-1)
= ∑ n(n-1) C(n-2,k-2) + ∑ n C(n-1,k-1)
= n(n-1) ∑ C(n-2,k-2) + n ∑ C(n-1,k-1)
= n(n-1) 2^n-2 + n 2^n-1
Now put n=28 , so the answer is
28 * 27 * 2^26 + 28 * 2^27
= 2^27 (14*27 +28)
= 406 * 2^27

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