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

We are given the following:

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

Each child must get at least 4 balloons and 1 pencil. We are tasked with finding how many ways we can distribute these items among the children while meeting these conditions.

Step 1: Distribute the Minimum Required Balloons and Pencils

For the Balloons: \[ \text{Each child gets 4 balloons. So, for 3 children: } 3 \times 4 = 12 \text{ balloons are given.} \] This leaves: \[ 15 - 12 = 3 \text{ balloons to be distributed.} \] Now, we use the formula for distributing \( n \) identical objects among \( r \) people/groups: \[ \binom{n+r-1}{r-1} \] Here, \( n = 3 \) (remaining balloons) and \( r = 3 \) (children). The number of ways to distribute 3 identical balloons among 3 children is: \[ \binom{3+3-1}{3-1} = \binom{5}{2} = \frac{5!}{2!3!} = 10 \]

Step 2: Distribute the Pencils

For the Pencils: \[ \text{Each child gets 1 pencil. So, for 3 children: } 3 \times 1 = 3 \text{ pencils are given.} \] This leaves: \[ 6 - 3 = 3 \text{ pencils to be distributed.} \] Again, using the same formula, for \( n = 3 \) pencils and \( r = 3 \) children, the number of ways to distribute 3 identical pencils among 3 children is: \[ \binom{5}{2} = 10 \]

Step 3: Distribute the Erasers

For the Erasers: \[ \text{There are 3 identical erasers and 3 children. The number of ways to distribute 3 identical erasers among 3 children is:} \] Using the formula for \( n = 3 \) erasers and \( r = 3 \) children: \[ \binom{5}{2} = 10 \]

Step 4: Calculate the Total Number of Ways

The total number of ways to distribute the items is the product of all the individual ways: \[ \text{Total} = 10 \times 10 \times 10 = 1000 \]

Final Answer:

There are \( \boxed{1000} \) ways to distribute the items among the children while satisfying the given conditions.

Answer 2

We are given:

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

These are to be distributed among 3 children such that:

• Each child gets at least 4 balloons
• Each child gets at least 1 pencil

 

Step 1: Minimum Allocation

Give each child 4 balloons and 1 pencil:
Total balloons given = \( 3 \times 4 = 12 \)
Total pencils given = \( 3 \times 1 = 3 \)

Step 2: Remaining Items

• Remaining balloons = \( 15 - 12 = 3 \)
• Remaining pencils = \( 6 - 3 = 3 \)
• Remaining erasers = 3 (no restriction on erasers)

Step 3: Distribute Remaining Items

We now need to distribute 3 identical balloons, 3 identical pencils, and 3 identical erasers among 3 children with no restriction.

The number of ways to distribute \( n \) identical items among \( r \) distinct recipients is given by: \[ \binom{n + r - 1}{r - 1} \] So, for each type: \[ \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10 \text{ ways} \]

Step 4: Multiply All Independent Distributions

Since the distributions are independent, total number of ways: \[ 10 \times 10 \times 10 = \boxed{1000} \]

✅ Final Answer: 1000 ways