Question

Prove the generalized version of the basic counting principle.

Short answer

Proof by mathematical induction. Use the basic principle of counting proven in the book.

Step-by-step solution

Given Information.

The generalized version of the basic counting principle.

Explanation of the given statement.

The Generalized Basic Principle of Counting,

If $r$experiments that are to be performed are such that the first one may result in any of $n_1$the possible outcomes, and if for each of these $n_1$possible outcomes.

There are$n_2$possible outcomes of the second experiment, and if for each of the possible outcomes of the first two experiments there are $n_3$possible outcomes of the third experiment, and if, .., then there is a total of$n_1·n_2·\dots ·n_r$possible outcomes of the$r$experiments.

Explanation.

Proof by mathematical induction

For $r=1,$only one experiment is performed and with $n_1$outcomes, and trivially there are $n_1$outcomes of this one experiment.

Let's say that the generalized principle of counting is true for specific$r\in \mathrm{\mathbb{N}}$experiments. $\left(1\right)$

If $r+1$experiments are performed and the first experiment can result in $n_1$outcomes, and for each outcome of the first experiment, the second experiment can result in $n_2$outcomes and ... and if for each outcome of the first$r$experiments, the $r+1.$experiment can result in $n_{r+1}$results:

By statement$\left(1\right)$, the first $r$experiments can end in any of the $n_1·n_2·\dots ·n_r$outcomes

By the basic principle of counting the first $r$and $r+1$., the experiment can jointly end in$\left(n_1·n_2·\dots ·n_r\right)·n_{r+1}$outcomes.

This experiment can, by the associative property of multiplication result in:

$n_1·n_2·\dots ·n_r·n_{r+1}$possible outcomes.

The principle of mathematical induction on localid="1652770742935" $\mathrm{\mathbb{N}}$the generalized principle of counting is true for every$r\in \mathrm{\mathbb{N}}$.