Question

If the number of ways in which 14 identical toys can be distributed among three boys so that each one gets at least one toy and no two boys get equal number of toys is m, then find the value of $\left(\frac{m}{20}\right)$.

Answer 1

Answer: 3

Le the boys get $a,b$ and $c$ toys respectively $a+b+c=14$,
$a+b+c=14$
give one to each
$a+b+c=11$
Using beggar method
Total ways $={}^{13}{C}_2=78$
now subtract following cases
$1,1,12$
$2,2,10$
$3,3,8$

Ans: $78-18=60=m$
$\therefore \frac{m}{20}=3$