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Question
If a and b are coefficients of xn in the expansions of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, then write the relation between a and b.
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Solution
\[\text{ Coefficient of } x^n \text{ in the expansion} (1 + x )^{2n} =^{2n}{}{C}_n = a\]
\[\text{ Coefficient of } x^n \text{ in the expansion} (1 + x )^{2n - 1} = ^{2n - 1}{}{C}_n = b\]
\[\text{ Now, we have:} \]
\[ ^{2n}{}{C}_n = \frac{2n!}{n! . n!} = \frac{2n(2n - 1)!}{n\left( n - 1 \right)! n!} . . . \left( 1 \right)\]
\[ \text{ and } ^{2n - 1}{}{C}_n = \frac{(2n - 1)!}{n!(n - 1)!} . . . \left( 2 \right)\]
\[\text{ Dividing equation } \left( 1 \right) \text{ by } \left( 2 \right), \text{ we get } \]
\[ \Rightarrow \frac{^{2n}{}{C}_n}{^{2n - 1}{}{C}_n} = \frac{2n(2n - 1)! n! (n - 1)!}{n\left( n - 1 \right)! n! (2n - 1)!}\]
\[ \Rightarrow \frac{a}{b} = 2\]
\[ \Rightarrow a = 2b\]
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