Answer

Correct Answer : d ) 84

In the given question, it is given:

There are 20 identical balloons to be distributed among 4 children such that:

a. No children get 0 balloons.

b. Every child gets an even number of balloons.

So, a child may get 2, 4, 6.... balloons

We can write even numbers as a product of 2 and an integer.

Now let us suppose that the four children get 2p, 2q, 2r, and 2s balloons.

2p + 2q + 2r + 2s = 20

p + q + r + s = 10

Now we need to find a combination of four positive integers which adds up to 10.

As p, q, r and s should be 1 or greater than 1.

Let us suppose we have 10 identical balls that we need to distribute among four people such that each person receives some.

Let us distribute 1 ball each to p, q, r and s. Now we need to distribute 6 balls among four people such that anyone can get any number of balls.

We know that,

The number of ways of distributing n identical things among r different groups so that each group gets 0 or more things = ^n+r-1C_r-1

Here, we get the total number of ways as ^6+4-1C_4-1 = ⁹C₃ = 84.

Hence, we have 84 ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons.

Therefore, (d) is the correct answer.