$^{(k - 1)} C_{(k - 1)} +^{k} C_{(k - 1)} +^{(k + 1)} C_{(k - 1)} + . . . . . . . +^{(n + k - 2)} C_{(k - 1)}$ $= ?$

^{(n + k)} C_{k}

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^{(n + k + 1)} C_{k}

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^{(n + k)} C_{k - 1}

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^{(n + k - 1)} C_{k}

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Solution

The correct option is D $^{(n + k - 1)} C_{k}$
$^{k - 1} C_{k} +^{k} C_{k - 1} - ------------- - + . . . . +^{n + k - 2} C_{k - 1}$
$=^{k + 1} C_{k} +^{k + 1} C_{k - 1} - ---------------- - + . . . . +^{n + k - 2} C_{k - 1}$
$=^{k + 2} C_{k} + . . . . . . . . . . . . . . . +^{n + k - 2} C_{k - 1} =^{n + k - 1} C_{k}$

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