Question

In how many ways can $20$ oranges be given to four children where each should get at least one orange?

• A. $869$
• B. $969$
• C. $973$
• D. None of these

Answer 1

The correct option is B $969$

The problem is equivalent to number of non-negative integral solutions of $x_1+x_2+...+x_r=n$ is ${}^{n+r-1}{C}_{r-1}$.

Since every student should get atleast 1 orange
After giving 1 orange to each student, 16 oranges are left to be distributed
Thus the problem is equivalent to finding the number of integral solutions for

$x_1+x_2+x_3+x_4=16$
and $n=16;r=4$
no. of ways = no. of integral solution = ${}^{n+r-1}{C}_{r-1}={}^{19}{C}_3=969$