Counting questions are about arrangements, selections and the number of ways of meeting a condition, and many end in a probability. The reports show a clear pattern: students know the formulae and lose marks on repeated letters, missing scenarios and adding where they should multiply. This guide collects what Cambridge examiners say students get wrong, based on recent reports.
Summarised in our own words from a reading of the examiner reports.
Mistakes that cost marks
- Adding instead of multiplying. Adding the combinations for the separate selections instead of multiplying them is a common error in the reports, and one report suggests that the "and / or" idea is not well embedded. The right way is to multiply for "and", and then add the totals for the different scenarios ("or").
- Repeated letters. Not dividing by the right factorial (such as 3! for three Ts), dividing every term by 3! when only some arrangements had repeats, or removing the repeats for one colour only. Another group assumed that repeated letters could be identified (Cs and Os in one question, 80 the most common wrong total).
- Repeated letters in probability. For a probability, a different approach is needed in the denominator: the report says the repeated letters should be considered identifiable to find the total number of outcomes. Omitting a repeated colour from the denominator was another common error.
- Not giving a probability. Several reports record final answers such as 7C1 × 5C3 = 70 when a probability was requested.
- Missing scenarios. Leaving out one of the groupings, such as the As in the group of 3 and the Ns in the group of 2, the case where neither Jane nor Kate was present, or the scenario 4 – 2 – 1. In another case, only the outcome for 4 letters between the Es was found, and not for 5, 6 and 7.
- Impossible scenarios. Including (5 men, 0 women) or (1 man, 4 women), which did not meet the conditions.
- Order matters, or does not. Not realising that two people can be arranged in two ways (not multiplying by 2), that men and women can swap positions, or that T and C could be reversed. In the other direction, students used permutations when combinations were needed, or thought the order of the Ns mattered and multiplied by 6P2.
- Extra terms. Introducing additional terms for selecting letters that had already been determined (85 was a common wrong value).
- Blocks. For a block of two letters with four letters between them, not realising that there are four positions for the block. In one question, students forgot that if Mr Lizo goes first his family also goes first, so only three families could change order.
- Misreading availability. Using 40 or 39 members available to travel when the conditions for two people removed some options.
- Rounding an exact value to 3 significant figures.
What examiners advise
- List the possible scenarios in a table, in a logical order, and work out the number of ways for each. Reports say good solutions did this, and many used simple diagrams for the arrangements.
- Two valid strategies exist for many "together" questions: subtract the arrangements where the condition fails from the total, or build the arrangement directly. Choose one and be consistent.
- Read the question carefully to see whether a probability is wanted, as many candidates did not.
- Leave exact answers exact.
Habits that help
- For each scenario, write the selections as a product of combinations, then add the products.
- Ask yourself, for every set of objects: does the order matter? Are any objects identical?
- When you finish, check whether your list of scenarios includes the case with none of a certain type.
Questions students ask
When do I add and when do I multiply combinations?
Multiply for separate selections that happen together ("and") and add for different scenarios ("or"). Adding combinations for separate selections is a common error. For each scenario write the selections as a product, then add the products.
How do I deal with repeated letters?
Divide by the factorial for each repeated letter, for example 3! for three Ts, but only where the repeat applies. Students divided every term by 3! when only some arrangements had repeats. For a probability, the denominator needs repeated letters treated as identifiable.
Why did I lose marks when the question asked for a probability?
Because the answer stopped at a count such as 7C1 × 5C3 = 70. Read the question to see if a probability is wanted, and divide by the total number of outcomes.
How do I make sure I include every scenario?
List the scenarios in a table, in a logical order, and work out the number of ways for each. Check the case with none of a certain type, and remove impossible ones such as (5 men, 0 women) if they break the conditions.
How do I solve "together" questions?
Two valid strategies: subtract the arrangements where the condition fails from the total, or build the arrangement directly. Choose one and be consistent. Ask for every set of objects whether order matters and whether any are identical.
Practice this topic
Ready to try some? Work through past paper questions on this topic one at a time. Attempt each question fully first, and open the mark scheme only afterwards.