Question

There are 3 ways to go from A to B, 2 ways to go from B to C and 1 way to go from A to C. In how many ways can a person travel from A to C?

Answer 1

1. Express the event of travelling from A to C as a compound statement using "AND" and "OR" as logical connectors.
2. Basic Principle of Counting: If there are m ways for happening of an event A, and corresponding to each possibility there are n ways for happening of event B, then the total number of different possible ways for happening of events A and B are:
   Either event A alone OR event B alone: $m+n$ .
   Both event A AND event B together: $m\times n$

Complete step by step solution:
It is given that there are 3 ways to go from A to B, 2 ways to go from B to C and 1 way to go from A to C.
Let $n\left( E \right)$ denote the number of ways of happening of an event E. Then, the given information can be summarized as follows:
$n\left( A\to B \right)=3$
$n\left( B\to C \right)=2$
$n\left( A\to C \right)=1$
The event of travelling from A to C can be written as a compound statement as follows:
[(Go from A to B) AND (Go from B to C)] OR [Go from A to C]
Using the Basic Principle of Counting, the above statement is equivalent to:
$\left[ n\left( A\to B \right)\times n\left( B\to C \right) \right]+\left[ n\left( A\to C \right) \right]$
Substituting the values, we get:
$=\left[ 3\times 2 \right]+\left[ 1 \right]$
$=6+1$
$=7$
Hence, there are a total of 7 ways for a person to travel from A to C. This is shown in the graph below:

The 7 ways of travelling from A to C, are:
1. $x_1, y_1$ 2. $x_1, y_2$
3. $x_2, y_1$ 4. $x_2, y_2$
5. $x_3, y_1$ 6. $x_3, y_2$
7. $z_1$

Note:

1. Mutually exclusive events are two events which cannot occur at the same time.
   These events are connected by logical "OR".
2. Independent events are those where one event remains unaffected by the occurrence of the other event.
   These events are connected by logical "AND".
3. Mutually exclusive events are necessarily also dependent events because one's existence depends on the other's non-existence.