Counting questions in Additional Math ask for arrangements, selections and the number of ways of meeting a condition. The reports show that the same choices cause the errors every year: permutations or combinations, multiply or add, and which cases to include. 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
- Permutations instead of combinations (and the reverse). The most common error in several questions. The reports also mention using combinations where permutations or simple factorials were appropriate, and using factorials in the product rule for counting.
- Adding instead of multiplying. Summing combinations that should have been multiplied (12 from a sum rather than the product), and, in a question with three cases, forming a sum of three sums instead of three products. Reports say that students who made little progress often did not know whether to multiply or add, and did not attempt a diagram to decide.
- Wrong cases. Omitting one or more of the cases, repeating a product already included, including an extra case (such as 4 teachers, 4 students and 2 others), and not separating the problem into distinct, mutually exclusive cases. Students who subtracted the cases to be discarded from all possible cases sometimes omitted one of those cases.
- Selecting from what remains. Not allowing for the fact that once three people have been chosen from twelve, only nine are left to choose from.
- Choosing independently of a condition. Choosing girls and boys independently of the requirement that the total is 6.
- Digit and arrangement problems. Assuming that a number ending in 5 could also start with 5, that 8 could be used at both the beginning and the end (answer 2520), counting an incorrect number of possibilities for the first digit, and treating 1 as a prime number. Others rearranged the remaining 4 books and then doubled (4! × 4! × 2).
- Complementary counting. Not multiplying 8! by 2 before subtracting it from 9!, or subtracting 2 × 7! from 9!, and adding 3 to 72 instead of subtracting.
- Notation. Writing two permutation or combination terms side by side with no sign showing that they are multiplied is not sufficient for a method mark if the final answer is wrong.
- Equations in n. Using (n − 5)! rather than (n + 1 − 5)! in a denominator, omitting n or one of the (n + 1) factors, and treating a result as an equation incorrectly. Reports say candidates who expanded factorials too fully often became entangled in algebra, and did less well than those who used concise factorial manipulation.
- Impossible answers. Giving an answer larger than the number of all possible arrangements of the digits from 0 to 9, which should have shown an error.
What examiners advise
- Decide whether order matters (permutation) or not (combination), and say so.
- Multiply for "and," add for "or," and draw a diagram or list the cases to see which applies.
- List the cases so that they do not overlap, and check that none is missing.
- Set out the solution so that an examiner can follow the process.
- Use the mark allocation and the answer space as a guide to how much work is expected.
- Manipulate factorials concisely in equations.
Habits that help
- Write the cases as a short list, with the number of ways for each one, before you add them.
- For a number question, write a box for each digit and put the number of choices in it.
- Check that your answer is smaller than the total number of arrangements without restrictions.
Questions students ask
How do I decide between permutations and combinations?
Ask whether order matters. Permutation if it does, combination if it does not. Using the wrong one is the most common error in several questions.
When do I multiply and when do I add?
Multiply for "and" and add for "or". Draw a diagram or list the cases to see which applies. Writing two terms side by side with no multiplication sign is not sufficient for a method mark if the final answer is wrong.
How do I make sure I have the right cases?
List the cases so that they do not overlap and none is missing. Students omitted cases, repeated a product or included an extra case. Remember that once three people are chosen from twelve, only nine remain.
How do I deal with digit problems?
Read the conditions carefully. Students assumed a number ending in 5 could also start with 5, or used 8 at both the beginning and the end. For complementary counting, multiply by the right factor before subtracting from the total.
How do I solve equations with factorials in n?
Manipulate factorials concisely and take care with brackets: (n − 5)! is different from (n + 1 − 5)! in a denominator. An answer larger than the number of all possible arrangements should signal an error.
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.