The correct option is A ^n+m+1C_n+1 Given #160; ∑ _ r=0 ^ m ^ n+r C_ n =^ n C_ n +^ n+1 C_ n +^ n+2 C_ n +^ n+3 C_ n .....^ n+m C_ n =^ n+ ...

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Combination

∑r=0m n+rCn=

Question

$m \sum r = 0^{n + r} C_{n} =$

^{n + m + 1} C_{n + 1}

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^{n + m + 2} C_{n}

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^{n + m + 3} C_{n + 1}

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none of these

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Solution

The correct option is A $^{n + m + 1} C_{n + 1}$
Given $\sum_{r = 0}^{m}^{n + r} C_{n}$
$=^{n} C_{n} +^{n + 1} C_{n} +^{n + 2} C_{n} +^{n + 3} C_{n} . . . . .^{n + m} C_{n}$
$=^{n + 1} C_{n + 1} +^{n + 1} C_{n} +^{n + 2} C_{n} +^{n + 3} C_{n} . . . . .^{n + m} C_{n}$ ( $∵^{n} C_{n} =^{n + 1} C_{n + 1}$ )
$=^{n + 2} C_{n + 1} +^{n + 2} C_{n} +^{n + 3} C_{n} . . . . .^{n + m} C_{n}$
( $∵^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ )
$=^{n + 3} C_{n + 1} +^{n + 3} C_{n} + . . . . . .^{n + m} C_{n}$
Continuing this ,
$=^{n + m} C_{n + 1} +^{n + m} C_{n}$
$=^{n + m + 1} C_{n + 1}$

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Similar questions

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