Question

A man wants to buy ' $m$ ' mangoes in ' $n$ ' different varieties, mangoes of the same variety being identical and they are available in abundance. Number of different ways he can plan his purchases, if he has to buy atleast two mangoes of the same variety is

• A. ${}^{m-2n}{C}_{n-1}$
• B. ${}^{m+n-3}{C}_{m-1}$
• C. ${}^{m+n-1}{C}_m-{}^n{C}_m$
• D. $\frac{{}^{m+n-1}{P}_{n-1}-{}^n{P}_m}{m!}$

Answer 1

Answer: (C)

Treat each of $V_1{V}_2\dots \dots \dots V_n$ varieties as $n$ beggars.
Total ways when $m$ mangoes can be taken without any restriction $=m+n-1C_m$
Now ${}^{m+n-1}{C}_m=$ All of same variety $+(m-1)$ of same variety and 1 diff. $+(m-2)$ of same variety and 2 diff. $+\dots \dots ..+$ all ' $m$ ' of diff. variety
At least two mangoes of the same variety $($ say $x)=$ Total - All $m$ of different variety
$={}^{m+n-1}{C}_m-{}^n{C}_m\Rightarrow (C)$