JEE Main 2022 (Online) 28th June Evening ShiftQuestion 1
The number of ways to distribute $30$ identical candies among four children $C_1,C_2,C_3$ and $C_4$ so that $C_2$ receives atleast $4$ and atmost $7$ candies, $C_3$ receives atleast $2$ and atmost $6$ candies, is equal to :
Permutation and Combination
Solution
Concept. Fix $x_2$ and $x_3$; then $x_1+x_4=30-x_2-x_3$ has $31-x_2-x_3$ solutions. Formulas used. (1) $x_1+x_4=t$ ($t\ge0$) has $t+1$ non-negative solutionsStep 1 - Pairs. $x_2\in\{4,..,7\}$ (4 values), $x_3\in\{2,..,6\}$ (5 values): 20 pairs. Always $t\ge30-13=17\ge0$. Step 2 - Sum. $\sum(31-x_2-x_3)=20\cdot31-\sum x_2-\sum x_3$. $\sum x_2=5(4+5+6+7)=110$, $\sum x_3=4(2+3+4+5+6)=80$. Step 3 - Value. $620-110-80=430$. Answer: D