Suppose there are 6 non-stop flights from Chennai to Mumbai in the morning and 4 non-stop flights from Mumbai to Goa in the evening. In how many ways can one fly from Chennai to Goa via Mumbai using these flights in a day?
Flights: Calculating Ways to Travel from Chennai to Goa via Mumbai
This problem asks us to find the total number of ways to travel from Chennai to Goa by taking a stopover in Mumbai. We are given the number of non-stop flights for each leg of the journey:
- Chennai to Mumbai: 6 non-stop flights in the morning.
- Mumbai to Goa: 4 non-stop flights in the evening.
To complete the journey from Chennai to Goa via Mumbai, one must first choose a flight from Chennai to Mumbai and then choose a flight from Mumbai to Goa.
Understanding the Counting Principle
When we have a sequence of events, and the number of ways for each event to occur is known and independent of the other events, we use the Multiplication Principle to find the total number of ways the sequence can occur. In this case, choosing a flight from Chennai to Mumbai is the first event, and choosing a flight from Mumbai to Goa is the second event.
The events are independent because the choice of flight from Chennai to Mumbai does not affect the choice of flight from Mumbai to Goa.
Applying the Multiplication Principle
Let $N_1$ be the number of ways to fly from Chennai to Mumbai, and $N_2$ be the number of ways to fly from Mumbai to Goa.
- $N_1 = 6$ (number of flights from Chennai to Mumbai)
- $N_2 = 4$ (number of flights from Mumbai to Goa)
The total number of ways to fly from Chennai to Goa via Mumbai is the product of the number of ways for each leg of the journey:
Total ways = $N_1 \times N_2$
Total ways = $6 \times 4$
Total ways = $24$
Result
There are 24 different ways one can fly from Chennai to Goa via Mumbai using the given flights.
We can also represent this with a simple structure:
| Journey Segment | Number of Options |
|---|---|
| Chennai to Mumbai | 6 |
| Mumbai to Goa | 4 |
| Total Ways (Chennai → Mumbai → Goa) | $6 \times 4 = 24$ |
Thus, the total number of ways to complete the travel from Chennai to Goa via Mumbai is 24.
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