Question

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

• A. 403
• B. 384
• C. 429
• D. 455

Hint:

Stars and bars method: For $x_1+...+x_r=n, x_i \ge 1$, ways = $\binom{n-1}{r-1}$.

Answer 1

The Correct Option is D

The oranges are identical and the children are distinct. Each child must receive at least one orange. The number of ways to distribute $n$ identical objects into $r$ distinct groups, with no group empty, is given by: \[ \binom{n-1}{r-1} \] Here, \[ n = 16, \quad r = 4 \] \[ \text{Number of ways} = \binom{15}{3} \] \[ = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = 5 \times 7 \times 13 = 455 \] \[ \therefore \text{the required number of ways is } \boxed{455}. \]