In a class there are 20 boys and 15 girls. The teacher wants to select either a boy or a girl to represent the

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Q. In a class there are 20 boys and 15 girls. The teacher wants to select either a boy or a girl to represent the class in a competition. In how many ways can this be done.

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Concept understanding — Fundamental Principles of Counting

The Fundamental Principle of Counting

Imagine you're getting dressed. You have 3 shirts (red, blue, green) and 2 pairs of pants (black, white). How many different outfits can you make?

You could list them: red-black, red-white, blue-black, blue-white, green-black, green-white. That's 6 outfits.

Notice something: 3 shirts × 2 pants = 6 outfits. That's not a coincidence — it's the entire idea.

The Intuition: A Fork in the Road

Think of each choice as a fork in a path. First you pick a shirt: 3 options. For each shirt, you then pick pants: 2 options. So the total number of paths through the whole process is 3 × 2 = 6.

This works no matter how many stages you add. If you also had 2 pairs of shoes, the total outfits become 3 × 2 × 2 = 12. Each new choice multiplies the total.

Note

The key insight: you multiply, not add. Addition would mean you choose either a shirt or pants, not both. Multiplication means you choose one of each, in sequence.

The Precise Statement

Fundamental Principle of Counting (Multiplication Principle)

If an event can happen in $m$ ways, and after it happens a second event can happen in $n$ ways, then the two events together can happen in $m \times n$ ways.

More generally: if a task consists of $k$ steps, and step 1 can be done in $n_1$ ways, step 2 in $n_2$ ways, ..., step $k$ in $n_k$ ways, then the total number of ways to complete the entire task is:

n_1 \times n_2 \times \cdots \times n_k

Watch out

This principle only works when the number of ways for each step does not depend on which choice was made earlier. If picking a red shirt somehow limits your pants options, you cannot simply multiply — you'd need to count more carefully.

A Simple Example

How many 2-digit numbers can you form using the digits 1, 2, 3, 4, 5 if repetition is allowed?

Step 1: Choose the tens digit — 5 options (1,2,3,4,5)
Step 2: Choose the units digit — 5 options (1,2,3,4,5)

Total = $5 \times 5 = 25$ numbers.

If repetition were not allowed:

Step 1: 5 options
Step 2: only 4 options (since one digit is already used)

Total = $5 \times 4 = 20$ numbers.

Tip

When repetition is not allowed, the number of options decreases by 1 at each step. When repetition is allowed, the number stays the same.

Why This Matters

This principle is the foundation of all counting problems in combinatorics. Permutations, combinations, arrangements — every one of them is just a special case of this multiplication idea, with some extra rules about repetition and order.

The rule is simple: when choices happen in sequence, multiply the number of options at each step.

Exactly one student — a boy or a girl — is selected, so the two mutually exclusive groups are added.

=

Number of boys $= 20$ .
Number of girls $= 15$ .
Selecting "either a boy or a girl" means exactly one student from the combined, non-overlapping groups.
Total ways $= 20+15$ .
$20+15 = 35$ .

✓Final answer

35

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Q.In a class there are 20 boys and 15 girls. The teacher wants to select either a boy or a girl to represent the class in a competition. In how many ways can this be done.

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