Question

How many different ways can $15$ candy bars be distributed between Ram, Shyam, Ghanshyam and Balram, if Ram can not have more than $5$ candy bars and Shyam must have at least two ? Assume all candy bars to be alike.

Answer 1

Distribute $15$ candies among Ram$\left(R\right)+$Shyam$\left(S\right)+$Ghanshyam$\left(G\right)+$Balram$\left(B\right)$ with condition given:
$R+S+G+B=15$ and $R\le 15,S\ge 2$
After giving $2$ to Shyam, remaining candies $=15-2=13$
Now distribute $13$ candies in $R,S,G,B$ in $\frac{(13+4-1)!}{\left(13\right)!3!}={}^{16}{C}_3$ ways
In ${}^{16}{C}_3$ ways, we have to remove undesirable ways$:$ $R>S\Rightarrow R\ge 6$
Give at least $6$ to $R$ and $2$ to $S$ and distribute remaining between $R,S,G,B$ in $\frac{(7+4-1)!}{\left(7\right)!(4-1)!}={}^{10}{C}_3$ ways.
${}^{10}{C}_3$ are undesirable cases.
$\therefore$ desired way $={}^{16}{C}_3-{}^{10}{C}_3=440$