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Question: The value of \(\sum_{r = 1}^{n + 1}\left( \sum_{k = 1}^{n}{kC_{r - 1}} \right)\) where r,k, n ĪN is ...

The value of ∑r=1n+1(∑k=1nkCr−1)\sum_{r = 1}^{n + 1}\left( \sum_{k = 1}^{n}{kC_{r - 1}} \right) where r,k, n ĪN is equal to

A

2n+1

B

2n+1 –1

C

2n+1 –2

D

None of these

Answer

2n+1 –2

Explanation

Solution

∑r=1n+1(∑k=1nkCr−1)\sum_{r = 1}^{n + 1}\left( \sum_{k = 1}^{n}{kC_{r - 1}} \right)= ∑r=1n+1(∑k=1n(k+1Cr−kCr))\sum_{r = 1}^{n + 1}\left( \sum_{k = 1}^{n}\left( k + 1C_{r} -^{k}C_{r} \right) \right)

= ∑r=1n+1(n+1Cr−1Cr)\sum_{r = 1}^{n + 1}\left( n + 1C_{r} -^{1}C_{r} \right) = 2n+1 – 2