Step 1: Understanding the Question:
This problem falls under the category of combinatorics and basic counting principles. We are tasked with finding the total number of distinct ways to distribute three specific scholarships (one for Mathematics, one for Chemistry, and one for Physics) among a group of available candidates. The question implies that each scholarship is a separate event and that we need to select one winner for each subject from the respective pools of candidates. Identifying whether the events are independent is the first step in solving such counting problems.
Step 2: Key Formulas and approach:
The primary tool used here is the Multiplication Principle (also known as the Fundamental Counting Principle). This principle states that if there are $n$ ways to perform one action and $m$ ways to perform another independent action, then there are $n \times m$ ways to perform both actions together. For this question, since the awarding of the Mathematics scholarship does not affect the awarding of the Chemistry or Physics scholarships, we simply need to find the product of the number of choices for each category.
Step 3: Detailed Explanation:
We identify the number of options available for each independent event:
Event 1: Selecting a winner for the Mathematics scholarship. There are 3 candidates available, so there are 3 possible outcomes.
Event 2: Selecting a winner for the Chemistry scholarship. There are 5 candidates available, so there are 5 possible outcomes.
Event 3: Selecting a winner for the Physics scholarship. There are 4 candidates available, so there are 4 possible outcomes.
To find the total number of ways all three scholarships can be awarded simultaneously, we apply the multiplication rule:
Total ways = (Number of Math choices) $\times$ (Number of Chemistry choices) $\times$ (Number of Physics choices).
Calculation: $3 \times 5 \times 4$.
First, $3 \times 5 = 15$.
Then, $15 \times 4 = 60$.
Thus, there are 60 distinct combinations of winners possible for these awards.
Step 4: Final Answer:
The total number of ways the scholarships can be awarded is 60, making (B) the correct option.