The ratio of the coefficients of x

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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