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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2018 · Mathematics · Binomial Theorem

If the coefficients of ^ m and ^ n in the expansion of (1+a)^ m+n...

If the coefficients of \mathrmam\mathrma^{\mathrm{m}} and \mathrman\mathrma^{\mathrm{n}} in the expansion of (1+a)m+n(1+a)^{\mathrm{m}+\mathrm{n}} are α\alpha and β\beta, then which one of the following is correct?

Options

  1. Aα=2β\alpha=2 \beta
  2. Bα=β\alpha=\beta
  3. C2α=β2 \alpha=\beta
  4. Dα=(m+n)β\alpha=(\mathrm{m}+\mathrm{n}) \beta

Correct answer

B. α=β\alpha=\beta

Step-by-step solution

(1+a)m+nα= coefficient of am=m+nCmβ= coefficient of an=m+nCn\begin{aligned} & (1+a)^{m+n} \\ & \alpha=\text{ coefficient of } a^m={ }^{m+n} C_m \\ & \beta=\text{ coefficient of } a^n={ }^{m+n} C_n \end{aligned} We know, nCr=nCn−r{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{n}-\mathrm{r}}∴β=m+nCn=(m+nm+n−n)=m+nCm=α∴α=β\begin{aligned} & \therefore \beta={ }^{m+n} C_n=\binom{m+n}{m+n-n}={ }^{m+n} C_m=\alpha \\ & \therefore \alpha=\beta \end{aligned}$$
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