Q72 / 160 · TS EAMCET 2017 · Mathematics

If the coefficients of $(2 r+1)^{\text {th }}$ term and $(r+1)^{\text {th }}$ term in the expansion of $(1+x)^{42}$ are equal then $r$ can be

  • A
    12
  • Correct answer
    B
    14
  • C
    16
  • D
    20

Solution

We have, $(1+x)^{42}$ $$ \begin{gathered} T_{2 r+1}={ }^{42} C_{2 r} x^{2 r} \\ T_{r+1}={ }^{42} C_r x^r \end{gathered} $$ Coefficient of $(2 r+1)^{\text {th }}$ and $(r+1)^{\text {th }}$ term are equal $$ \begin{aligned} & \therefore \quad{ }^{42} C_{2 r}={ }^{42} C_r \\ & \therefore \quad 2 r+r=42 \quad\left[\because{ }^n C_x={ }^n C_y \Rightarrow x+y=n\right] \\ & \Rightarrow \quad r=14 \\ & \end{aligned} $$

Hence, the correct answer is option: 'B'.

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