The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

The number of ways in which 16 oranges can be distributed to four children such that each child gets at least one orange

Q. The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is

A. 384

B. 403

C. 429

D. 455

Correct Answer: 455

Explanation

The oranges are identical and the children are distinct, so this is a problem of distribution of identical objects.

Let the number of oranges received by the four children be:

x₁, x₂, x₃ and x₄

Given condition:

x₁ + x₂ + x₃ + x₄ = 16

Since each child must get at least one orange:

x₁ ≥ 1, x₂ ≥ 1, x₃ ≥ 1, x₄ ≥ 1

Convert this into non-negative integers by substitution:

x₁ = y₁ + 1
x₂ = y₂ + 1
x₃ = y₃ + 1
x₄ = y₄ + 1

Substitute into the equation:

(y₁ + 1) + (y₂ + 1) + (y₃ + 1) + (y₄ + 1) = 16

y₁ + y₂ + y₃ + y₄ = 12

Now, the number of non-negative integer solutions of this equation is given by the stars and bars method.

Number of ways = C(12 + 4 − 1, 4 − 1)

= C(15, 3)

= (15 × 14 × 13) ÷ 6

= 455

Hence, the required number of ways is 455.