Distributing Identical Balloons

Let the number of balloons received by the four children be $x_1, x_2, x_3, x_4$. We are given:

1. $x_1 + x_2 + x_3 + x_4 = 20$
2. Each child gets some balloons ($x_i > 0$)
3. No child gets an odd number of balloons ($x_i$ is even and positive, so $x_i \in \{2, 4, 6, \dots\}$)

Let $x_i = 2k_i$ where $k_i \ge 1$ is an integer.

Substitute $x_i = 2k_i$ into the equation: $2k_1 + 2k_2 + 2k_3 + 2k_4 = 20$ $k_1 + k_2 + k_3 + k_4 = 10$

Where $k_i \ge 1$.

The number of positive integer solutions to $k_1 + k_2 + k_3 + k_4 = 10$ is given by: $\binom{10 - 1}{4 - 1} = \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$