The number of ways 16 oranges distributed to 4 children, each get… — Solution | JEE Mathematics

MCQEasyJEE 2026Applications of P&C

The number of ways $16$ oranges distributed to $4$ children, each gets at least one.

Answer

Correct answer:D

Standard Method

Given: $16$ oranges are to be distributed among $4$ children, and each child gets at least one orange.

Find: The number of possible distributions.

The oranges are identical and the children are distinct. This is a standard stars and bars problem with positive integral solutions.

For

x_1 + x_2 + x_3 + x_4 = 16, \qquad x_i \ge 1

the number of ways is

\binom{16-1}{4-1}

So,

\binom{15}{3}

Now,

\binom{15}{3} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1}

= 5 \times 7 \times 13

= 455

Therefore, the required number of ways is $455$ . The correct option is D.

Using combinations of children instead of distributions of identical oranges. This is wrong because the objects are identical and the problem asks for positive integer solutions. Use the stars and bars formula for $x_1+x_2+x_3+x_4=16$ with $x_i\ge 1$ .
Using $\binom{16}{4}$ or $\binom{16}{3}$ directly. This is wrong because at least one orange must go to each child first, so the correct count is $\binom{15}{3}$ , not a direct selection from $16$ oranges.
Applying the non-negative formula $\binom{n+r-1}{r-1}$ without adjusting the condition. That formula is for $x_i\ge 0$ , but here the condition is $x_i\ge 1$ . For positive distributions, use $\binom{n-1}{r-1}$ .

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