Question

$15$ identical jewels are to be distributed between $P,Q,R$ and $S$. Find the number of ways in which each one of them gets at least one.

• A. $364$
• B. $360$
• C. $332$
• D. $352$

Answer 1

The correct option is A $364$

We require $P,Q,R,S\ge 1$
$P+Q+R+S=15$
$\Rightarrow (P-1)+(Q-1)+(R-1)+(S-1)=11$
$\Rightarrow$ Everyone has at least $1$ jewel.

Let $(P-1),(Q-1),(R-1),(S-1)$ be $x,y,z,w$ respectively.
$\Rightarrow x+y+z+w=11\text{and}x,y,z,w\ge 0$

Now, we have to distribute $11$ jewels to $4$ individuals.
Required no. of ways are ${}^{n+r-1}{C}_{r-1}={}^{14}{C}_3$ $=364$ ways