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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2022 · Mathematics · Binomial Theorem

What is the value of...

What is the value of C(51,21)−C(51,22)+C(51,23)−C(51,24)+C(51,25)−C(51,26)+C(51,27)−C(51,28)+C(51,29)−C(51,30)C(51,21)-C(51,22)+C(51,23)-C(51,24)+C(51,25)-C(51,26)+C(51,27)-C(51,28)+C(51,29)-C(51,30) ?

Options

  1. AC(51,25)C(51,25)
  2. BC(51,27)C(51,27)
  3. CC(51,51)−C(51,0)C(51,51)-C(51,0)
  4. DC(51,25)−C(51,27)C(51,25)-C(51,27)

Correct answer

C. C(51,51)−C(51,0)C(51,51)-C(51,0)

Step-by-step solution

(5121)−(5122)+(5123)−(5124)+(5125)−(5126)+(5127)−(5128)+(5129)−51C30=(5151−21)−(5151−22)+(5151−23)−(5151−24)+(5151−25)−(5126)+(5127)−(5128)+(5129)−(5130)[∵nCr=nCn−1]=(5130)−(5129)−(5128)−(5127)+(5126)−(5126)+(5127)−(5128)+(5129)−(5130)=0Now, (5151)−51C0=1−1=0\begin{aligned} & \binom{51}{21}-\binom{51}{22}+\binom{51}{23}-\binom{51}{24}+\binom{51}{25}-\binom{51}{26}+\binom{51}{27}-\binom{51}{28} \\ & +\binom{51}{29}-{ }^{51} C^{30} \\ & =\binom{51}{51-21}-\binom{51}{51-22}+\binom{51}{51-23}-\binom{51}{51-24}+\binom{51}{51-25}- \\ & \binom{51}{26}+\binom{51}{27}-\binom{51}{28}+\binom{51}{29}-\binom{51}{30} \quad\left[\because{ }^{\mathrm{n}} C_{\mathrm{r}}={ }^{\mathrm{n}} C_{\mathrm{n}-1}\right] \\ & =\binom{51}{30}-\binom{51}{29}-\binom{51}{28}-\binom{51}{27}+\binom{51}{26}-\binom{51}{26}+\binom{51}{27}-\binom{51}{28} \\ & +\binom{51}{29}-\binom{51}{30} \\ & =0 \\ & \text{Now, }\binom{51}{51}-{ }^{51} C_0=1-1=0\end{aligned}
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