Question

Number of ways in which $25$ identical pens can be distributed among Keshav, Madhav, Mukund and Radhika such that at least $1,2,3$ and $4$ pens are given to Keshav, Madhav, Mukund and Radhika respectively, is

• A. ${}^{18}{C}_4$
• B. ${}^{28}{C}_3$
• C. ${}^{24}{C}_3$
• D. ${}^{18}{C}_3$

Answer 1

The correct option is D ${}^{18}{C}_3$

First of all select $1+2+3+4=10$ pens
out of $25$ identical pens and distribute them as desired.
It can happen only in one way.
Now let $x_1,x_2,x_3,x_4$ pens are given to them respectively.
$($ here,$x_1,x_2,x_3,x_4\ge 0)$
As now any one can get any number of pens.
$\therefore$ non-negative integral solution of $x_1+x_2+x_3+x_4=15$
will be the number of ways of ${}^{15+4-1}{C_{4-1}=}^{18}{C}_3$