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Sum of Coefficients of All Terms

The value of ...

Question

The value of the expression ${^{k - 1} C}_{k - 1} + {^{k} C}_{k - 1} + . . . . {^{n + k - 2} C}_{k - 1}$ is given by :

{^{n + k - 1} C}_{k - 1}

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{^{n + k - 1} C}_{k}

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{^{n + k} C}_{k}

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None of these

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Solution

The correct option is B ${^{n + k - 1} C}_{k}$
We know,
$n \sum m = k^{m} C_{r} =^{n + 1} C_{r + 1}$
According to the question,
$m = n + k - 2$
$r = k - 1$
Sum of series is given by
$=^{m + 1} C_{k + 1}$ $=^{n + k - 1} C_{k}$

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