====== 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.

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