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:
We need to find the total number of ways to award scholarships. Based on the options, the question implies awarding one scholarship for each of the three distinct subjects (Mathematics, Chemistry, and Physics).


Step 2: Key Formula or Approach:
We use the Fundamental Principle of Counting (Multiplication Rule). If one event can occur in \(m\) ways and a second independent event can occur in \(n\) ways, then the two events can occur together in \(m \times n\) ways.


Step 3: Detailed Explanation:
Number of ways to award the Mathematics scholarship = 3 (since there are 3 candidates)
Number of ways to award the Chemistry scholarship = 5 (since there are 5 candidates)
Number of ways to award the Physics scholarship = 4 (since there are 4 candidates)
Since the selections for each subject are independent of one another, the total number of ways to award all three scholarships is the product of the individual choices: \[ \text{Total ways} = 3 \times 5 \times 4 \] \[ \text{Total ways} = 60 \]

Step 4: Final Answer:
The scholarships can be awarded in 60 ways.

Answer 2

This problem asks for the total number of ways to award three separate scholarships, one each in Mathematics, Chemistry, and Physics, to candidates drawn from three separate, non-overlapping pools. Since a person's eligibility for the Mathematics scholarship has nothing to do with who wins Chemistry or Physics, we can build up the total using a decision-tree argument rather than jumping straight to the multiplication formula.

Picture the process as three independent stages. At the first stage, we choose who wins the Mathematics scholarship: there are 3 candidates, so this stage branches into 3 separate paths.

At the second stage, for each of those 3 branches, we choose who wins the Chemistry scholarship independently, out of the 5 Chemistry candidates. So each of the 3 Mathematics branches splits into 5 Chemistry branches, giving \(3\times5=15\) branches so far.

At the third stage, for each of those 15 branches, we choose who wins the Physics scholarship out of the 4 Physics candidates, splitting each of the 15 branches into 4 further branches.

The total number of complete branches, and hence complete ways to award all three scholarships, is:

\[ 15\times4=60 \]

Checking the other options: 12 would only arise from multiplying just two of the three subject counts (like \(3\times4\)) and forgetting Chemistry entirely; 20 would arise similarly from an incomplete pairing (like \(5\times4\)) that ignores Mathematics. Since all three independent stages must be accounted for, 60 is the only total consistent with all three scholarships being awarded.

Therefore, the correct answer is 60.