The multiplication rule counts outcomes formed by choices made together or in sequence. If there are n₁ options for the first choice, n₂ for the second, and so on, multiply them: total outcomes = n₁ × n₂ × … × nᵢ. Use this form when each choice leaves the stated number of options available at the other stages. section 2.2.1
Count outcomes before calculating probability
Two six-sided dice produce 6 × 6 = 36 ordered outcomes: (3, 2) and (2, 3) are different. To find the probability that the first die shows 1, 2, or 3 and the second shows 4, 5, or 6, count 3 × 3 = 9 successful outcomes. The probability is 9/36 = 25%. section 2.2.1 section 3.2.3
The same reasoning gives 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000 plates with three letters followed by three numerals. When every position has the same options and reuse is allowed, exponentiation shortens the product: a three-digit lock has 10³ = 1,000 settings. If reuse is not allowed, later positions have fewer options; do not keep multiplying by the original count. section 2.2.1 section 2.2.3
>>> num_permutations = n ** r
>>> print(f'Number of permutations with replacement: {num_permutations}')
Number of permutations with replacement: 125
The Python listing computes 5³ as `n ** r` and prints 125 permutations with replacement: five available choices at each of three positions. section 2.2.3
Choose the rule to match the question. Multiply counts for combined independent choices; add counts for mutually exclusive alternatives, such as flipping a coin *or* rolling a die. For independent events, a related probability rule multiplies their individual probabilities—for example, the probabilities of rolling a 4 on each of two dice. section 2.2.2 section 3.2
