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Coefficients of Terms Equidistant from Beginning and End

The coefficie...

Question

The coefficient of $x^{m}$ in $(1 + x)^{m} + (1 + x)^{m + 1} + \dots + (1 + x)^{n},$ where $m, n \in N$ and $m < n$ , is

A

^{n} C_{m}

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B

^{n} C_{m + 1}

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C

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

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D

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

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Solution

The correct option is C $^{n + 1} C_{m + 1}$
The coefficient of $x^{m}$ in
$(1 + x)^{m} + (1 + x)^{m + 1} + (1 + x)^{m + 2} + \dots + (1 + x)^{n} =^{m} C_{m} +^{m + 1} C_{m} +^{m + 2} C_{m} + \dots +^{n} C_{m} =^{m + 1} C_{m + 1} +^{m + 1} C_{m} +^{m + 2} C_{m} + \dots +^{n} C_{m} (∵^{m} C_{m} =^{m + 1} C_{m + 1} = 1) =^{n + 1} C_{m + 1} (∵^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1})$

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The coefficient of $x^{m}$ in $(1 + x)^{m} + (1 + x)^{m + 1} + . . . . . + (1 + x)^{n}$ , m £ n is

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y = \frac{1}{1 + x^{n - m} + x^{p - m}} + \frac{1}{1 + x^{m - n} + x^{p - n}} + \frac{1}{1 + x^{m - p} + x^{n - p}}, t h e n \frac{d y}{d x}

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Coefficients of Terms Equidistant from Beginning and End

Standard XI Mathematics

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