show that the number of ways if selecting $n -$ object out of $3 n -$ objects, $n$ of which are like and rest different is $2^{2 n - 1} + \frac{(2 n)!}{2 {(n!)}^{2}}$

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show that the number of ways if selecting n−object out of 3n−objects, n of which are like and rest differnt is $2^{2 n - 1} + \frac{(2 n)!}{2 {(n!)}^{2}}$

solution:
$\begin{matrix} A c c o r d i n g t o q u e s : w e h a v e 3 n - - - o b j e c t s i n w h i c h s e l e c t^{'} n^{'} o b j e c t s, a n d, o b j e c t s^{'} 3 n^{'} h a s^{'} n^{'} i s i d e n t i c a l a n d^{'} 2 n^{'}, d i f f e r e n t o b j e c t s . N o w, n u m b e r o f w a y s : n, i d e n t i c a l o b j e c t s - --------------------- - : 0, 1, 2.....3...... n a n d, 2 n, d i f f e r e n t o b j e c t - --------------------- - : n, n - 1, n - 2, . . . . . . n - 3.......0 t h e n t h e n o . o f w a y s a n d a d d i n g t h e n o . o f w a y s :^{2 n} c_{0} +^{2 n} c_{1} +^{2 n} c_{2} + . . . . . . .^{2 n} c_{3} + . . . . . .^{2 n} c_{n} [w e k n o w t h a t :^{2 n} c_{0} +^{2 n} c_{1} +^{2 n} c_{2} + . . . . .^{2 n} c_{n} + . . . . .^{2 n} c_{n + 1} . . . .^{2 n} c_{2 n} = 2^{2 n}] A g a i n, \Rightarrow^{2 n} c_{n + 1} +^{2 n} c_{n + 2} . . . . . .^{2 n} c_{2 n}, w e c a n a l s o w r i t e a s : \Rightarrow^{2 n} c_{2 n - (n + 1)} +^{2 n} c_{2 n (n + 2)} . . . . . . .^{2 n} c_{2 n - 2 n} | (^{n} c_{r} =^{n} c_{n - r}) \Rightarrow^{2 n} c_{n - 1} +^{2 n} c_{n - 2} . . . . . . .^{2 n} c_{0} A g a i n w e w r i t e : \Rightarrow^{2 n} c_{0} +^{2 n} c_{1} +^{2 n} c_{2} + . . . . .^{2 n} c_{n - 1} +^{2 n} c_{n} +^{2 n} c_{n + 1} . . . . . .^{2 n} c_{2 n} = 2^{2 n} a d d i n g^{2 n} c_{n} i n t o R . H . S \Rightarrow^{2 n} c_{0} +^{2 n} c_{1} +^{2 n} c_{2} + . . . . .^{2 n} c_{n - 1} +^{2 n} c_{n} - -------------------------------------- - +^{2 n} c_{n + 1} . . . . . .^{2 n} c_{2 n} - ----------------- - = 2^{2 n} +^{2 n} c_{n} s u p p o s e w e v a l u e d t h e e q u^{n} : s + s = 2^{2 n} +^{2 n} c_{n} \Rightarrow 2 s = 2^{2 n} +^{2 n} c_{n} \Rightarrow 2 s = 2^{2 n} + \frac{(2 n)!}{n! n!} ∴ s = 2^{2 n - 1} + \frac{(2 n)!}{2 {(n!)}^{2}} (p r o v e) \end{matrix}$

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