Count linked choices with a product

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Count linked choices with a product

If every first choice permits the same number of second choices, an outcome table has equal-length rows. Multiplying the number of rows by the number of entries per row gives the total. This is the product rule for counting.

The rule depends on the restrictions. A two-digit code with repetition permitted has ten choices in each position, giving 100 codes including leading zeros. If repetition is forbidden, the second position has only nine choices. If a code must be a two-digit number, the first digit cannot be zero, changing the count again.

Worked example: How many outfits use 3 shirts, 2 trousers and 2 jackets?

Each shirt permits 2 × 2 = 4 trouser/jacket combinations.

There are 3 shirt choices, each with four outcomes.

Answer: 12 outfits.

Check your understanding: How many ordered two-letter codes use A, B, C without repeated letters?

Answer: 6.

Why: There are three first choices and two remaining second choices: 3 × 2.