JEE Main 2024 · Mathematics · Permutations and Combinations · medium

The number of ways to distribute 10 identical sweets among 4 children so that every child gets at least 1 sweet and no child gets more than 4 sweets…

Single correct option

The number of ways to distribute

10

identical sweets among

4

children so that every child gets at least

1

sweet and no child gets more than

4

sweets is

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Answer: (C) $44$

Let

x_i

be the sweets of child

i

and put

y_i=x_i-1

, so

0\le y_i\le3

and

y_1+y_2+y_3+y_4=6

. Without the upper bound the count is

\binom{9}{3}=84

. Now remove the cases where some

y_i\ge4

: write

y_i=4+z

, so

z+(\text{other three})=2

, which has

\binom{5}{3}=10

solutions, and there are

4

choices of the child. Two children cannot both have

y\ge4

since

8>6

, so there is no overlap to add back. The answer is

84-4\cdot10=44

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