Distribution of balloons, pencils, and erasers

First, give each of the 3 children 4 balloons and 1 pencil.

• Remaining balloons = $15 - 3 \times 4 = 3$
• Remaining pencils = $6 - 3 \times 1 = 3$
• Remaining erasers = 3

Number of ways to distribute $N$ identical items among 3 children is given by $\binom{N + 3 - 1}{3 - 1} = \binom{N + 2}{2}$.

• Ways for 3 balloons: $\binom{3+2}{2} = \binom{5}{2} = 10$
• Ways for 3 pencils: $\binom{3+2}{2} = \binom{5}{2} = 10$
• Ways for 3 erasers: $\binom{3+2}{2} = \binom{5}{2} = 10$

Total ways = $10 \times 10 \times 10 = 1000$.