Fundamental counting principle drills

Count outcomes using the multiplication (fundamental counting) principle.

Free, timed SHSAT practice sets on fundamental counting principle— every question includes a full worked explanation, and there's no account required.

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How to solve counting principle problems

If you make several choices in a row, multiply the number of options for each choice.

To count an inclusive range like "2 through 9", use last \(-\) first + 1: \(9 - 2 + 1 = 8\) digits. The digits 0 through 9 are 10 digits, not 9.

Common traps: adding the options instead of multiplying, leaving out one category, and the off-by-one count of an inclusive range.

Worked examples

Example 1.
A store offers 3 desserts, 4 appetizers, 6 main courses. If a customer picks exactly one from each category, how many different meal combinations are possible?
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Answer:
B

By the counting principle, multiply the number of choices in each category: 3 \(\times\) 4 \(\times\) 6 = 72. (Adding the counts instead of multiplying (\(3 + 4 + 6\)) gives 13. Leaving out the main courses (\(3 \times 4\)) gives 12. Multiplying the first two counts and then adding the third (\(3 \times 4 + 6\)) gives 18.)

Example 2.
A security code uses 2 digits, each chosen from 0 through 7, and digits MAY repeat. How many different codes are possible?
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Answer:
A

The digits 0 through 7, counted inclusively, give 7 \(-\) 0 + 1 = 8 choices per digit. \(8^{2} = 64\). (Counting \(7 - 0 = 7\) digits instead of 8 (\(7^{2}\)) gives 49. Multiplying 8 by 2 instead of raising it to the power 2 gives 16. Not letting digits repeat (\(8 \times 7\)) gives 56.)