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

Find the rati...

Question

Find the ratio of the coefficients of $x^{n}$ and $x^{q}$ in the expansion of $(1 + x)^{p + q} .$

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Solution

Coefficients of $x^{n}$ in the expansion of $(1 + x)^{2 n} i s^{2 n} C_{n} = a$

Coefficients of $x^{n}$ in the expansion of $(1 + x)^{2 n - 1} i s^{2 n - 1} C_{n} = b$

Now,

$\frac{a}{b} = \frac{^{2 n} C_{n}}{^{2 n - 1} C_{n}} = \frac{\frac{(2 n)!}{n! n!}}{\frac{(2 n - 1)!}{n! (n - 1)!}} = \frac{2 n}{n} = 2$

Hence, $\frac{a}{b} = 2.$

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

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