Question

The number of ways of distributing 15 identical balloons, 6 identical pencils and 3 identical erasers among 3 children, such that each child gets at least four balloons and one pencil, is

Answer 1

Given:

• 15 identical balloons
• 6 identical pencils
• 3 identical erasers
• 3 children

Each child must receive a minimum of 4 balloons and 1 pencil. The objective is to determine the number of ways to distribute these items under these constraints.

Step 1: Distribute Minimum Required Balloons

Minimum balloons per child: \( 4 \). Total distributed initially: \( 3 \times 4 = 12 \). Remaining balloons: \( 15 - 12 = 3 \). Distributing \( n=3 \) identical balloons among \( r=3 \) children: \( \binom{n+r-1}{r-1} = \binom{3+3-1}{3-1} = \binom{5}{2} = 10 \) ways.

Step 2: Distribute Minimum Required Pencils

Minimum pencils per child: \( 1 \). Total distributed initially: \( 3 \times 1 = 3 \). Remaining pencils: \( 6 - 3 = 3 \). Distributing \( n=3 \) identical pencils among \( r=3 \) children: \( \binom{n+r-1}{r-1} = \binom{3+3-1}{3-1} = \binom{5}{2} = 10 \) ways.

Step 3: Distribute Erasers

Distributing \( n=3 \) identical erasers among \( r=3 \) children: \( \binom{n+r-1}{r-1} = \binom{3+3-1}{3-1} = \binom{5}{2} = 10 \) ways.

Step 4: Calculate Total Distribution Ways

Total ways are the product of ways for each item distribution: \( 10 \times 10 \times 10 = 1000 \).

Final Answer:

There are \( \boxed{1000} \) ways to distribute the items according to the specified conditions.