Question:
The ratio of the coefficients of xm and xn in the expansion of (1 + x)m + n is ___________.
Solution:
For (1 + x)m + n
Tr+1 = m+nCr xr
for coefficient of xm, r = m
i.e coefficient is m+nCm
and for coefficient of xn, r = n
i.e. coefficient is m+nCn
$\Rightarrow$ Ratio of coefficient of $x^{m}$ to $x^{n}$ is $\frac{m+n}{m+n} C_{m}=\frac{{ }^{m+n} C_{m+n-m}}{m+n C_{n}}$
$=\frac{{ }^{m+n} C_{n}}{{ }^{m+n} C_{n}}=1$
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