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

Consider a school community of m families, with of ni 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.

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2. Choose one of the children at random.

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

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.