Question

The number of ways of distributing $20$ identical fruits among $5$ people, so that no one receives less than $3$ fruits is

• A. $126$
• B. $180$
• C. $196$
• D. $216$

Answer 1

The correct option is A $126$

Let $x_1,x_2,x_3,x_4,x_5$ be the fruits recieved by five people then
$x_1+x_2+x_3+x_4+x_5=20,x_1,x_2,x_3,x_4,x_5\ge 3$

Let $X_1=x_1-3,X_2=x_2-3,X_3=x_3-3,X_4=x_4-3,X_5=x_5-3$
So,
$X_1+X_2+X_3+X_4+X_5=20-(3+3+3+3+3)\Rightarrow X_1+X_2+X_3+X_4+X_5=5,X_1,X_2,X_3,X_4,X_5\ge 0$

The number of non negative integral solutions
$={}^{5+5-1}{C}_{5-1}={}^9{C}_4=126$