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

Correct Answer: 1000

Distribution Problem: Balloons, Pencils, and Erasers

Given: 
- 15 identical balloons.  
- 6 identical pencils. 
- 3 identical erasers. 
- 3 children.

Each child must get at least 4 balloons and 1 pencil. First, let's distribute the minimum required balloons and pencils to each child:

For the Balloons:

Each child gets 4 balloons. So, for 3 children: \(3 \times 4 = 12\) balloons are given. We're left with \(15 - 12 = 3\) balloons to be distributed.

Now, let's use the formula for distributing \(n\) identical objects among \(r\) people/groups.

The formula is: \(\binom{n+r-1}{r-1}\) 
Where: 
- \(n\) = number of identical objects 
- \(r\) = number of groups/people.

Here, \(n = 3\) (remaining balloons) and \(r = 3\) (children).

Number of ways to distribute 3 identical balloons among 3 children: 
\(\binom{3+3-1}{3-1} = \binom{5}{2}\) 
\(= \frac{5!}{2!3!} = 10\)

For the Pencils:

Each child gets 1 pencil. So, for 3 children: \(3 \times 1 = 3\) pencils are given. We're left with \(6 - 3 = 3\) pencils to be distributed.

Using the formula again, for \(n = 3\) pencils among \(r = 3\) children:

Number of ways to distribute 3 identical pencils among 3 children = \(\binom{5}{2} = 10\)

For the Erasers:

There are 3 identical erasers and 3 children. So, using the formula for \(n = 3\) erasers and \(r = 3\) children:

Number of ways to distribute 3 identical erasers among 3 children = \(\binom{5}{2} = 10\)

Total Number of Ways:

Now, the total number of ways is the product of all the individual ways: 
\(Total = 10 \, (\text{for balloons}) \times 10 \, (\text{for pencils}) \times 10 \, (\text{for erasers})\) 
\(Total = 1000\)

So, there are 1000 ways to distribute the items among the children satisfying the given conditions.