Question 3 Exercise 6.3

Solutions of Question 3 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Find $n$, when $^{2 n} C_3:^n C_2=36: 3$.

$$\begin{align} & { }^{2 n} C_3:{ }^n C_2=36: 3 . \\ & \Rightarrow \dfrac{(2 n) !}{(2 n-3) ! 3 !} \times \dfrac{(n-2) ! 2 !}{n !}=12 \\ & \Rightarrow \dfrac{2 n(2 n-1)(2 n-2)(2 n-3) !}{(2 n-3) ! 3 !}\times\dfrac{(n-2) ! 2 !}{n(n-1)(n-2) !}=12 \\ & \Rightarrow \dfrac{4 n(2 n-1)(n-1)}{3} \cdot \dfrac{1}{n(n-1)}=12 \\ & \Rightarrow 2 n-1=9 \\ & \Rightarrow 2 n=10 \\ & \Rightarrow n=5 \end{align}$$

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