The number of ways to distribute 30 identical candies among four children \(\mathrm{C}_{1}, \mathrm{C}_{2}, \mathrm{C}_{3}\) and \(\mathrm{C}_{4}\) so that \(\mathrm{C}_{2}\) receives atleast 4 and atmost 7 candies, \(\mathrm{C}_{3}\) receives atleast 2 and atmost 6 candies, is equal to

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Permutations and Combinations

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

Answer: (D)

Let

C_{2}

receive

x_{2}

candies where

x_{2} \in [4, 7]

and

C_{3}

receive

x_{3}

candies where

x_{3} \in [2, 6]

.

After allocating the minimum to

C_{2}

and

C_{3}

, we need to distribute

30 - 4 - 2 = 24

candies among four children with constraints.

Using generating functions or inclusion-exclusion principle:
The coefficient of

x^{24}

(1 + x + \dots)^{2} (1 + x + x^{2} + x^{3}) (1 + x + \dots + x^{4})

.

This equals:

^{27} C_{3} -^{23} C_{3} -^{22} C_{3} +^{18} C_{3}

= 2925 - 1771 - 1540 + 816 = 430

JEE Main · 2022 · 28 Jun 2022 · Shift 2 · Single correct option