Permutations and Combinations – Distribution of Objects – JEE Main 23 Jan 2026 Shift 2

Solution:

Let $x_1, x_2, x_3, x_4$ be the number of oranges received by the four children.
$$ x_1 + x_2 + x_3 + x_4 = 16 $$

Since each child gets at least one orange ($x_i \ge 1$), we use the formula for positive integral solutions:
$$ ^{n-1}C_{r-1} $$
where $n = 16$ and $r = 4$.

$$ ^{16-1}C_{4-1} = ^{15}C_3 $$

Calculation:
$$ ^{15}C_3 = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} $$

$$ = 5 \times 7 \times 13 $$

$$ = 35 \times 13 = 455 $$

Ans. (4)