# Question

Consider a school community of m families, with ni of them having i children,

Consider the following two methods for choosing a child:

1. Choose one of the m families at random and then randomly choose a child from that family.

2. Choose one of the

ini children at random.

Show that method 1 is more likely than method 2 to result in the choice of a firstborn child.

In solving this problem, you will need to show that

To do so, multiply the sums and show that, for all pairs i, j, the coefficient of the term ninj is greater in the expression on the left than in the one on the right.

Consider the following two methods for choosing a child:

1. Choose one of the m families at random and then randomly choose a child from that family.

2. Choose one of the

ini children at random.

Show that method 1 is more likely than method 2 to result in the choice of a firstborn child.

In solving this problem, you will need to show that

To do so, multiply the sums and show that, for all pairs i, j, the coefficient of the term ninj is greater in the expression on the left than in the one on the right.

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