Permutations and combinations — common mistakes

Look for keywords like 'arrange', 'order', 'line up', 'sequence', or specific roles (President, 1st place) to identify a permutation problem. Remember that $^n P_n = n!$.

Always read the question carefully to identify all constraints. If a problem seems too complex, break it down into smaller, manageable parts. For example, 'at least 2 men' might mean 'exactly 2 men' OR 'exactly 3 men' OR... Add the results of these separate cases.

The key difference is order. In permutations, the order of the items is important (e.g., ABC is different from BCA). In combinations, the order does not matter (e.g., the group {A, B, C} is the same as {B, C, A}).

Look for keywords. 'Arrange', 'order', 'sequence', 'line up', or specific named positions (like President, Treasurer) suggest a permutation (nPr). 'Choose', 'select', 'pick', 'group', or 'committee' suggest a combination (nCr).

This is often best handled with the 'slot' or 'gap' method. First, place the items that do not have restrictions. This creates 'gaps' (including at the ends) where the restricted items can be placed. For example, to separate A and B in CDE, first arrange CDE (3! ways). This creates 4 gaps (_ C _ D _ E _). Then choose 2 of these 4 gaps for A and B ($^4P_2$ ways).

Yes, your calculator has functions for nPr and nCr, which are essential for saving time. However, you must show your method in the exam, for example by writing '$^8C_3$' or '$\frac{8!}{3!5!}$' before writing the final answer. For complex problems involving multiple steps (like choosing men AND women for a committee), you need to show how you are combining these calculations (e.g., $^6C_2 \times ^5C_2$).