ISI2014-DCG-72

Question

The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j  \: \: {}^kC_j$ is equal to 

• A. $-1$
• B. $0$
• C. $1$
• D. $2^n$

Comments

may be I committed some mistake.

Will check again.

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okay. May be solved by writing inner summation as binomial expression.

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Yes!

Answer 1

From the binomial theorem,

$\displaystyle (1+x)^k=\sum_{j=0}^{k} {{}^{k}\textrm{C}_{j}}~x^j$

Putting $x=-1$ to equation above yields,

$\displaystyle (1-1)^k=\sum_{j=0}^{k} {{}^{k}\textrm{C}_{j}}(-1)^j\\ \displaystyle \Rightarrow \sum_{j=0}^{k} (-1)^j~{{}^{k}\textrm{C}_{j}}=0$

 

Now

$\displaystyle \sum_{k=1}^{n}(-1)^k~{{}^{n}\textrm{C}_{k}} \sum_{j=0}^{k} (-1)^j~{{}^{k}\textrm{C}_{j}}=\sum_{k=1}^{n}(-1)^k~{{}^{n}\textrm{C}_{k}} \times 0=0$

 

So the correct answer is B.