Multiplication principle: if one choice can be made in m ways and a second choice in n ways, the two together can be made in m × n ways. Draw a box for each position and write the number of choices in it.

Factorials: n! = n × (n − 1) × … × 2 × 1 is the number of ways to arrange n different objects in a row. By definition 0! = 1.

Worked example. How many 3-digit numbers can be made from the digits 1 to 6 if no digit is repeated? Boxes: 6 × 5 × 4 = 120. If repeats were allowed it would be 6 × 6 × 6 = 216.

A permutation is an arrangement where order matters. The number of ways to arrange r objects chosen from n different objects is [sup n]P[sup r] = [frac n!/(n − r)!].

If some objects are identical, divide by the factorial of each repeated group: the letters of a word with n letters, where one letter appears a times and another b times, can be arranged in n!a! × b! ways. For objects that must be together, glue them into one block, arrange the blocks, then arrange inside the block.

Worked example. In how many ways can a president and a secretary be chosen from 10 club members? Order matters (the roles differ): ¹⁰P² = 10 × 9 = 90.

A combination is a selection where order does not matter. The number of ways to choose r objects from n different objects is [sup n]C[sup r] = [frac n!/r! (n − r)!]. Each combination of r objects can be arranged in r! orders, so ⁿC^r = ⁿP^rr!.

Ask yourself: if I swap two of the chosen objects, do I get a different outcome? If yes, use permutations; if no, use combinations. For at least one questions, count everything and subtract the selections with none.

Worked example. How many ways can a team of 3 be chosen from 8 players? ⁸C³ = 8 × 7 × 63 × 2 × 1 = 3366 = 56. Compare with ⁸P³ = 336: each team is counted 3! = 6 times when order matters.