For selections, the order does not matter, so you use nCr. The extra work is in the conditions: someone must be in, someone must be out, or the group needs a minimum of one type.
This lesson follows counting arrangements with restrictions in the permutations and combinations chapter.
How do I handle a person who must be in or out?
A required person uses up one place, so both n and r fall by one. Choosing 5 from 9 students, with Ali required, means choosing 4 from the remaining 8: 8C4 = 70.
An excluded person only lowers n. Choosing 5 from 9 with Ali out means choosing 5 from 8: 8C5 = 56.
Worked example: at least two girls
A team of 5 is chosen from 5 boys and 4 girls. It must have at least 2 girls. How many teams are possible?
Method 1: case by case. List the allowed numbers of girls and add.
| Girls | Boys | Count |
|---|---|---|
| 2 | 3 | 4C2 × 5C3 = 6 × 10 = 60 |
| 3 | 2 | 4C3 × 5C2 = 4 × 10 = 40 |
| 4 | 1 | 4C4 × 5C1 = 1 × 5 = 5 |
The total is 60 + 40 + 5 = 105.
Method 2: complement. All teams: 9C5 = 126. Teams with fewer than 2 girls: 0 girls gives 1, and 1 girl gives 4 × 5C4 = 4 × 5 = 20. That is 21 unwanted teams.
126 − 21 = 105. The methods agree, which confirms the answer.
The mistake that costs marks
The tempting shortcut is to pick 2 girls first, then any 3 of the remaining 7 people: 4C2 × 7C3 = 6 × 35 = 210. This looks tidy but gives double the true answer.
The reason is that a team with 3 girls is counted 3 times, once for each pair of its girls that could be the “first 2 girls”. The next lesson explains this over-counting in detail.
Check yourself
A committee of 4 is chosen from 3 teachers and 4 students. It must include at least 1 teacher. How many committees are there?
Answer
Use the complement. All committees: 7C4 = 35.
Committees with no teacher: all 4 from students, 4C4 = 1.
The answer is 35 − 1 = 34.
Check by cases: 1 teacher gives 3 × 4C3 = 12, 2 teachers give 3C2 × 4C2 = 18, 3 teachers give 1 × 4C1 = 4. The sum is 12 + 18 + 4 = 34.
What to study next
Continue with distinguishing over-counting from under-counting to see why the shortcut above fails and how to test any count. The word-problem structure worksheet helps you mark the required and excluded members.
If at-least questions are where marks slip, see online one-to-one Additional Mathematics tuition.