Question 1

What is the main difference between permutations and combinations?

Accepted Answer

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}).

Question 2

How do I know whether to use nPr or nCr in an exam question?

Accepted Answer

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).

Question 3

What if items need to be separated, not together?

Accepted Answer

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).

Question 4

Can I just use the formulas on my calculator?

Accepted Answer

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., $^6C_2 	imes ^5C_2$).

Permutations and combinations

What you need to know

Arranging vs. Choosing