Counting principle

Counting principle questions are similar to probability questions, but instead of probabilities they are asking about possibilities. Often in ways such as: how many different possible combinations are there of several different groups of items or options, such as letters and numbers for a license plate or dice rolls from multiple pairs of dice (possibly with different sets of numbers).

The counting principle simply states that if there are x possible ways to do one thing and y possible ways to do another thing, then there are x • y possible ways of doing both things. So count and then multiply.

This can be thought of in a similar way to multiplying probabilities from above, but where the tree branch of options is more about the possibilities created by each branch rather than what proportion of them is successful. Basically only the denominators calculated for that tree matter, rather than the whole fractions.

Here is an example question from the December 2017 ACT:

Single digits 0 through 9 = 10 possible numbers that can be chosen

Letters of the alphabet = 26 possible numbers that can be chosen.

Single digits must be chosen for each of the first three slots and they can repeat. Therefore, the are 10 ways to fill slot #1, 10 ways to fill slot #2, and 10 ways to fill slot #3 = 10 • 10 • 10 = 1, 000 possibilities. This is how many possible ways the first three slots can be filled.

The last three slots require letters. There are 26 ways to fill each slot, and so there are 26 • 26 • 26 = 17, 576 possibilities. This is how many ways the last three slots can be filled.

How many ways can all 6 slots be filled= first three possibilities x last three possibilities = 1, 000  • 17, 576 = 17, 576, 000. Our answer therefore is K.