In the binomial expansion of $(1 + x)^{m + n}$ , prove that the coefficients of $x^{m}$ and $x^{n}$ are equal.

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Solution

In the expansion of $(1 + x)^{m + n},$
Coeffocient of $x^{m} =^{m + n} C_{n}$ and coefficient of $x^{n} =^{m + n} C_{n} .$
But $^{m + n} C_{m} =^{m + n} C_{n}$ ( $∵^{n} C_{r} =^{n} C_{n - r}$ )
Hence, coefficient of $x^{m}$ and $x^{n}$ are equal.

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