Counting Principles • Topic 3 of 3

Addition Principle

The addition principle handles alternatives: if a task can be completed by method A OR by method B, and the two methods share no outcome (mutually exclusive), the total number of ways is the sum. The trigger word is "OR". The non-negotiable condition is disjointness — the two cases must not overlap, or you will double-count. CAT loves problems that look like one task but split into clean cases: "a 3-digit number that is either divisible by 5 or has all even digits", or "the committee has exactly 2 men OR exactly 3 men". The standard CAT technique, called counting by cases, is to break a constrained problem into mutually exclusive sub-problems, count each by the multiplication principle, and add. When cases can overlap, switch to inclusion–exclusion (A + B − both); but if you choose your cases to be naturally exclusive, plain addition is enough and far faster.

✅ Solved examples

1. A committee of 3 is chosen from 5 people, but must include a specific person P or not — count both totals. How many committees with P plus how many without P?
With P: choose 2 more from remaining 4 = 6. Without P: choose 3 from 4 = 4. Either case (disjoint): 6 + 4 = 10 = total ⁵C₃. (Confirms addition over exclusive cases.)
2. How many integers from 1 to 50 are divisible by 5 OR by 7?
Divisible by 5: 10; by 7: 7; by 35: 1 (overlap). 10 + 7 − 1 = 16. (Cases overlap, so subtract the both-count.)
3. A 2-digit number is to be formed from {1,2,3,4} that is either even-ended or starts with 1. Count via exclusive cases.
Total 2-digit numbers (rep. allowed) = 16. Even-ended (units 2 or 4): 4 × 2 = 8. Odd-ended and starting with 1: units in {1,3} (2 ways), tens = 1: 1 × 2 = 2. These two cases are disjoint: 8 + 2 = 10.
4. In how many ways can you score exactly 4 from two dice using sums, counting ordered rolls?
Rolls summing to 4: (1,3),(2,2),(3,1). Three mutually exclusive outcomes ⇒ 3 ways.

✏️ Practice — try these, take hints as needed

1. How many integers from 1 to 60 are divisible by 4 or by 6?
Count multiples of 4 and of 6 separately.
Subtract multiples of 12 (overlap).
15 + 10 − 5.
20
2. A coin is tossed and a die is rolled. How many outcomes give a head OR a six? (count distinct outcomes)
Total outcomes = 12.
Heads: 6; a six: 2; both (head and six): 1.
6 + 2 − 1.
7
3. From a group of 4 boys and 3 girls, a team of 2 with all same gender. How many teams?
Two disjoint cases: both boys OR both girls.
⁴C₂ + ³C₂.
6 + 3.
9
4. How many 2-digit numbers (10–99) start with 3 or end with 7?
Start with 3: 10 numbers; end with 7: 9 numbers.
Overlap (37): 1.
10 + 9 − 1.
18
5. A snack is either a fruit (4 kinds) or a drink (3 kinds). How many ways to pick exactly one snack?
Pick a fruit OR pick a drink.
Mutually exclusive.
4 + 3.
7

📝 Topic test — 8 questions

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