Principles of counting · PEA515

The multiplication principle says that if one task can be done in m ways and an independent second task in n ways, the two together can be done in m x n ways. The addition principle says that if two tasks are alternatives that cannot both be chosen, there are m + n ways.

Situation	Count
Choose one of each from independent groups	Multiply the group sizes
Either this or that	Add
k-digit numbers from n digits, repetition allowed	`n^k`
k-digit numbers from n digits, no repetition	`n x (n-1) x ... x (n-k+1)`
Even numbers	Fill the restricted units place first
Grid paths with R right and U up moves	`C(R + U, R)`

Example: 6 shirts, 8 trousers and 2 pairs of shoes give 6 x 8 x 2 = 96 outfits. A journey with 3 + 4 routes from A to B and 2 + 3 + 1 routes from B to C has (3 + 4) x (2 + 3 + 1) = 42 ways: add alternatives within a leg, multiply across legs.

Exam trap: adding the numbers of choices instead of multiplying them is the most common error. Decide first whether repetition is allowed.

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Question 1 · Tier 3, Apply

A code has two letters chosen from the first 5 letters of the alphabet (repetition allowed), followed by 1 digit (0 to 9, repetition allowed). How many codes are possible?

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C. Letters: 5 x 5 ways; digits: 10^1. Total 250.

Why not A: Assumes the letters cannot repeat.

Why not B: Adds the letter choices.

Why not D: Adds the letter and digit counts instead of multiplying.

Question 2 · Tier 3, Apply

How many even 4-digit numbers can be formed using the digits 1 to 5 if no digit may be repeated?

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D. Fill the units place first in 2 ways, then the other 3 places in P(4, 3) ways: 48.

Why not A: Ignores the condition that the number is even.

Why not B: Assumes exactly half of the numbers are even.

Why not C: Allows repetition in the other places.

Key points

Try two questions