Question

There are 3 candidate for a Mathematics, 5 for chemistry and 4 for a Physics scholarship. In how many ways can the scholarship be awarded.

• A. 12
• B. 60
• C. 20
• D. none of these

Hint:

When events are independent and you need them all to occur ("AND" condition), multiply their respective possibilities. If it were a choice of awarding only ONE scholarship in total ("OR" condition), you would add them (\(3+5+4=12\)).

Answer 1

The Correct Option is B

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.

Answer 2

Think of every possible outcome as a labeled point in a 3-dimensional grid: one axis has 3 positions (the Mathematics candidates), a second axis has 5 positions (the Chemistry candidates), and a third axis has 4 positions (the Physics candidates). Every valid way of awarding all three scholarships corresponds to exactly one point in this grid, and every point corresponds to exactly one way of awarding the scholarships.

1. 12: This would be the size of a 2-dimensional slice of the grid, for instance \(3\times4\), which leaves out an entire axis (Chemistry) and undercounts the full 3-dimensional space of outcomes.
2. 60: This is the total number of points in the full 3-dimensional grid, obtained by multiplying the sizes of all three axes together: \(3\times5\times4=60\). Every one of these 60 points represents one distinct, complete way to award all three scholarships at once.
3. 20: This would come from another 2-dimensional slice, such as \(5\times4\), again leaving out an entire axis (Mathematics) and undercounting the space.
4. None of these: Since the 3-dimensional grid calculation gives a clean, well-defined answer of 60, there is no reason to fall back on this option.

Only the full three-axis grid, sized \(3\times5\times4\), accounts for every combination of winners across all three scholarships.

Therefore, the correct answer is 60.