Chapter 9, Problem 1CC

To determine

(a)

To define:

The Fundamental Counting Principle.

Explanation of Solution

Fundamental Counting Principle:

If there are $m$ number of events $E_1,E_2,E_3,\mathrm{........},E_m$ that can occur in $n_1,n_2,n_3,\mathrm{........},n_m$ different ways in order respectively then the events can occur in $n_1\cdot n_2\cdot n_3\cdot \mathrm{........}\cdot n_m$ ways.

To determine

(b)

To find:

The number of routes from $A$ to $C$ via $B$ if there are three roads from town $A$ to town $B$ and five roads from town $B$ to town $C$.

Answer to Problem 1CC

The number of routes from $A$ to $C$ is $15$

Explanation of Solution

Given information:

There are three roads from town $A$ to town $B$ and five roads from town $B$ to town $C$.

Concept used:

Fundamental Counting Principle:

If there are $m$ number of events $E_1,E_2,E_3,\mathrm{........},E_m$ that can occur in $n_1,n_2,n_3,\mathrm{........},n_m$ different ways in order respectively then the events can occur in

$n_1\cdot n_2\cdot n_3\cdot \mathrm{........}\cdot n_m$ ways.

Calculation:

Here it is given that there are three roads from town $A$ to town $B$ and five roads from town $B$ to town $C$.

So the number of events is

$2$ in which the first event can occur in $3$ ways and the second event can occur in $5$ ways.

Therefore by the fundamental counting principle there are $3\cdot 5=15$ number of routes from town $A$ to town $C$.

Conclusion:

Thus, the number of routes from $A$ to $C$ is $15$.