A person travels from Hyderabad to Goa and returns, but does not use the same bus for both journeys. If there are 25 buses available for each direction, how many ways can the round trip be made?
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In "round trip without repetition" problems, if there are $n$ options, the answer is always $n(n-1)$. If the same bus could be used, it would be $n^2$.
Step 1: Understanding the Concept:
This problem operates on the Fundamental Principle of Counting (Multiplication Principle). If one event can occur in $m$ ways and a second independent event can occur in $n$ ways, then the two events can occur in sequence in $m \times n$ ways. Step 2: Key Formula or Approach:
Identify the number of independent choices for the onward journey ($m$) and the return journey ($n$). The total number of ways is $m \times n$. Step 3: Detailed Explanation:
The round trip consists of two legs:
1. Onward Journey (Hyderabad to Goa): The person can choose any of the 25 available buses.
So, number of ways for the onward journey = 25.
2. Return Journey (Goa to Hyderabad): The condition states that the person {does not use the same bus} for both journeys. This means the specific bus taken for the onward journey is now excluded from the choices for the return trip.
Available buses for the return journey = $25 - 1 = 24$.
So, number of ways for the return journey = 24.
Applying the multiplication principle, the total number of ways to complete the round trip is:
\[ \text{Total ways} = 25 \times 24 \]
\[ 25 \times 24 = 25 \times (25 - 1) = 625 - 25 = 600 \] Step 4: Final Answer:
The round trip can be made in 600 ways.