Question

The number of 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, is

Answer 1

The correct answer is 1771

Answer 2

The generating function for each child can be represented as:
\((1 + x^2 + x^4 + ...)\)
Since no child can get an odd number of balloons, we are only interested in the even powers of x. Therefore, we can rewrite the generating function as:
\((1 + x^2 + x^4 + ...) = \frac{1}{(1 - x^2)}\)
Now, for four children, the generating function for the entire scenario is the product of the individual generating functions:
\((\frac{1}{(1 - x^2)})^4\)
To find the coefficient, using the binomial theorem:
\((\frac{1}{(1 - x^2)})^4 = C(4 + 20 - 1, 20)\times{x^{20}}\)
Simplifying further:
\(C(23, 20)\times{x^{20}} = (\frac{23!}{(20!\times{3!)}})\times{x^{20}} = 1771\times{x^{20}}\)
Therefore, the number of 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 is 1771