A counting method overcounts when the same final outcome can be produced by more than one sequence of choices, so it is counted more than once. To explain it, name one outcome that is counted more than once and say how many times. This reasoning skill completes permutations and combinations, and it comes after using a complement because that method is the correct alternative in the example below.
How do you test a proposed method?
- Read the method as a story of choices. What is chosen first, what second?
- Ask whether the order of choosing creates duplicates. If a group can be built in two different orders of choosing, it is counted twice.
- Find a specific outcome that appears more than once, and count how many times.
- Confirm with a small check or by working out the correct answer another way.
Worked example
A committee of 3 is chosen from 6 boys and 4 girls. A student proposes this method for “at least one girl”: choose 1 girl first (4 ways), then choose any 2 others from the remaining 9 people (9C2 = 36 ways). The answer is proposed as 4 × 36 = 144. Explain why this is wrong and find the correct answer.
Find a repeated outcome. Take the committee {Girl 1, Girl 2, Boy 1}. The method counts it once when Girl 1 is chosen first and Boy 1 and Girl 2 are the “others”, and again when Girl 2 is chosen first. So it is counted twice. A committee with 3 girls is counted 3 times, once for each girl who could be the “first”.
Count to check the 144.
| Girls in committee | Committees | Times counted | Contribution |
|---|---|---|---|
| 1 | 4 × 6C2 = 60 | 1 | 60 |
| 2 | 4C2 × 6 = 36 | 2 | 72 |
| 3 | 4C3 = 4 | 3 | 12 |
60 + 72 + 12 = 144. The table confirms the method’s answer and shows exactly where the extra comes from.
Correct answer: the actual number of committees is 60 + 36 + 4 = 100. By complement: 10C3 − 6C3 = 120 − 20 = 100. The two routes agree.
Explanation in a sentence: a committee with more than one girl is counted once for each girl who could have been chosen first, so the total is too large.
The mistake to watch for
A common slip is to accept the method because every step sounds reasonable.
Mistaken reasoning: “We need at least one girl, so we choose one girl first to make sure of it. Then the rest can be anyone. That covers every case.”
The logic makes sure a girl is present, but it does not make each committee appear only once. The correction is to test with a specific committee that has two girls and see that it appears twice.
Check yourself
1. A method for choosing a pair from 7 people picks a first person (7 ways) and then a second (6 ways), giving 42. Explain the error and give the correct number.
Show answer
The pair {Ali, Bala} arises as Ali then Bala and as Bala then Ali, so every pair is counted twice. Correct: 42 / 2 = 21, which is 7C2.
2. A committee of 4 is chosen from 5 men and 4 women. To find the number with at least 2 women, a student chooses 2 women (4C2 = 6 ways) and then any 2 from the other 7 people (21 ways), giving 126. Explain the error and find the correct number.
Show answer
A committee with 3 women is counted 3 times (any 2 of the 3 women can be the “chosen” pair), and a committee with 4 women is counted 6 times.
Correct number: 2 women and 2 men: 6 × 10 = 60. 3 women and 1 man: 4 × 5 = 20. 4 women: 1. Total = 81.
Check by complement: 9C4 − 5C4 − 4 × 5C3 = 126 − 5 − 40 = 81.
3. Five people sit in a row, and A and B must sit together. A student treats AB as one block and arranges 4 objects, giving 4! = 24. Is this too large or too small, and what is correct?
Show answer
It is too small. The block can be AB or BA, so the count must be multiplied by 2. Correct: 24 × 2 = 48.
Where this leads next
Work through the mixed practice set, where some questions ask you to judge a proposed method. You can also return to the module overview to see how the five lessons fit together.
Explaining why a method fails is a different skill from getting the answer, and it is hard to practise alone. Our online one-to-one Additional Mathematics tuition gives you a teacher to explain to and to check that your reasoning is complete.