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Standard XII

Mathematics

Index of (r+1)th Term from End When Counted from Beginning

If a and b ar...

Question

If a and b are the coefficients of xn in the expansion of $(1 + x)^{2 n}$ and $(1 + x)^{2 n - 1}$ respectively, find $\frac{a}{b}$

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Solution

Coefficient of $X^{p}$ in the expansion of $(1 + x)^{p + q} i s^{p + q} C_{p}$

Coefficient of $X^{q}$ in the expansion of $(1 + x)^{p + q} i s^{p + q} C_{q}$

Now,

$\frac{^{p + q} C_{p}}{^{p + q} C_{q}} = \frac{\frac{(p + q)!}{p! q!}}{\frac{(p + q)!}{q! p!}} = 1$

Hence, the ratio of the coefficients of $x^{n}$ and $x^{q}$ in the expansion of $(1 + x)^{p + q} i s 1 : 1.$

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