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Question
If a and b are the coefficients of xn in the expansion of \[\left( 1 + x \right)^{2n} \text{ and } \left( 1 + x \right)^{2n - 1}\] respectively, find \[\frac{a}{b}\]
Solution
Coefficients of xn in the expansion of \[\left( 1 + x \right)^{2n}\] is \[{}^{2n} C_n = a\] .
Coefficients of xn in the expansion of \[\left( 1 + x \right)^{2n - 1}\] is \[{}^{2n - 1} C_n = b\] .
Now,
\[\frac{a}{b} = \frac{{}^{2n} C_n}{{}^{2n - 1} C_n}\]
\[ = \frac{\frac{\left( 2n \right)!}{n!n!}}{\frac{\left( 2n - 1 \right)!}{n!\left( n - 1 \right)!}}\]
\[ = \frac{2n}{n}\]
\[ = 2\]
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