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Given that the binomial expansion term is ${\left( {1 + x} \right)^{m + n}}$

We have to prove that the coefficients of ${x^m}$ and ${x^n}$ are the same.

So, first let us find the coefficients of ${x^m}$ and ${x^n}$ in the expansion.

As we know that for any general binomial expansion term ${\left( {1 + x} \right)^k}$ the general coefficient of ${x^r}$ is given by \[{}^k{C_r}\] .

Using the above formula let us find the coefficients of ${x^m}$ and ${x^n}$ .

The coefficient of ${x^m}$ is \[{}^{m + n}{C_m}\] .

Similarly the coefficient of ${x^n}$ is \[{}^{m + n}{C_n}\] .

Now we have the coefficients of both the terms we have to prove they are equal.

As we know the general formula for combination term is:

\[{}^k{C_r} = {}^k{C_{k - r}}\]

Using the above formula let us manipulate the coefficient of ${x^n}$ . So, we have:

\[

\Rightarrow {}^{m + n}{C_n} = {}^{m + n}{C_{\left( {m + n} \right) - n}} \\

\Rightarrow {}^{m + n}{C_n} = {}^{m + n}{C_m} \\

\]

This is the same as the coefficient of ${x^m}$ .

Hence, the coefficient of ${x^m}$ and ${x^n}$ are equal.

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