Counting and Arrangements

Almost every counting question rests on one idea. If you can make a first choice in some number of ways, and then a second choice in some number of ways, multiply. Three shirts, then four pairs of pants, then two hats gives three times four times two, which is twenty-four outfits. Each shirt goes with every pair of pants, and each of those goes with every hat. But when a question splits into separate cases, joined by the word or, you add the cases instead. A code that is one letter or one digit has twenty-six plus ten options.

When one place has a rule, fill that place first. How many three-digit numbers can you make from the digits zero to six, with no digit repeated? The hundreds digit cannot be zero, so it has six choices. The tens digit can be zero, but it cannot repeat the hundreds digit, so it also has six choices out of the seven digits. The units digit has five left. Six times six times five is one hundred and eighty. If you fill the free places first, you lose track of whether zero has been used.

Five different books go on a shelf, in five factorial ways, which is one hundred and twenty. Now suppose the two maths books must be together. Glue them into one block. You now have four things to arrange, in four factorial ways, which is twenty-four. Inside the block, the two maths books can go either way round, so multiply by two, giving forty-eight. For never next to each other, it is much easier to count the opposite. Take the forty-eight together arrangements away from all one hundred and twenty, which leaves seventy-two.

Sometimes it is easier to count too many on purpose, then divide. To choose a team of three from six people, count ordered picks: six times five times four is one hundred and twenty. But each team has been counted once for every order of its three members, which is three factorial, six times. So there are twenty teams. Five people round a table: there are one hundred and twenty orders in a line, but each seating can be rotated five ways and still look the same, so twenty-four. The word LEVEL: one hundred and twenty orders, but the two Ls can swap and the two Es can swap, so divide by four, giving thirty.

When a question says at least one, count the opposite. A four-digit code uses the digits one to five, and repeats are allowed. How many codes contain at least one five? Every code: five choices in each of four places, six hundred and twenty-five. Codes with no five at all: four choices in each place, two hundred and fifty-six. Every other code has at least one five, so the answer is six hundred and twenty-five minus two hundred and fifty-six, which is three hundred and sixty-nine. Quiz and exercises now.