Choose the correct answer.

The coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^{2n - 1} are in the ratio.Hint: $^{2\text{n}}\text{C}_\text{n} : \ ^{2\text{n} - 1}\text{C}_\text{n}$

Question

Choose the correct answer. The coefficient of x n in the expansion of (1 + x) 2n and (1 + x) 2n - 1 are in the ratio. Hint: $^{2\text{n}}\text{C}_\text{n} : \ ^{2\text{n} - 1}\text{C}_\text{n}$

Aicharya · Accepted Answer

Solution: General Term $\text{T}_{\text{r}+1}=\ ^\text{n}\text{C}_\text{r}\text{x}^{\text{n}-\text{r}}\text{y}^\text{r}$ In the expansion of (1 + x) 2n , we get $\text{T}_{\text{r}+1}=\ ^{2\text{n}}\text{C}_\text{r}\text{x}^\text{r}$ To get the coefficient of x n , put r = n $\therefore$ Coefficient of $\text{x}^\text{n}=\ ^{\text{2n}}\text{C}_\text{n}$ In the expansion of $(1+\text{x})^{2\text{n}-1},$ we get $\text{T}_{\text{r}+1}=\ ^{2\text{n}-1}\text{C}_\text{r}\text{x}^\text{r}$ $\therefore$ Coefficient of $\text{x}^\text{n}\ \ \text{is}=\ ^{\text{2n}-1}\text{C}_{\text{n}-1}$ The required ratio is $\frac{^{\text{2n}}\text{C}_{\text{n}-1}}{^{\text{2n}-1}\text{C}_{\text{n}-1}}$ $=\frac{\frac{2\text{n}!}{\text{n}!(\text{n}!)}}{\frac{(2\text{n}-1)!}{(\text{n}-1)!(2\text{n}-1-\text{n}+1)!}}=\frac{\frac{2\text{n}!}{\text{n}!.\text{n}!}}{\frac{(2\text{n}-1)!}{(\text{n}-1)!(\text{n}!)}}$ $=\frac{2\text{n}!}{\text{n}!\text{n}!}\times\frac{(\text{n}-\text{n})!\cdot\text{n}!}{(2\text{n}-1)!}=\frac{2\text{n}(2\text{n}-1)!}{\text{n}!\text{n}(\text{n}-1)!}\times\frac{(\text{n}-1)!\cdot\text{n}!}{(2\text{n}-1)!}$ $=\frac{2}{1}=2:1$ Hence, the correct option is (d).

Choose the correct answer. The coefficient of x n in the expansion of (1 + x) 2n and (1 + x) 2n - 1 are in…

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