GCSE(H),

The *Product Rule* takes effect when more than one choice is made from a selection of options. The product rule consists of multiplying the number of selections, at each stage, together:

*number of permutations = n_{1} n_{2} n_{3}..*.

where *n*_{1} is the number of permutations for selection 1; *n*_{2} is the number of permutations for selection 2, and so on.

The number of permutations may change depending on a previous selection (as in a lottery, for example, where a drawn ball cannot be drawn again).

1. Six coloured stripes are being used as a background to a logo. Red and yellow, the corporate colours, must be included; a further 8 colours are available for the remaining stripes. How many permutations are there for the background if no colour may be used twice?

Answer: 50,400

Select red, the choice is position. There are 6 possible positions for red: *n*_{1} = 6

Select yellow. There are 5 remaining positions for yellow: *n*_{2} = 5

Select a remaining position. The choice is colour. There are 8 remaining colours, *n*_{3} = 8

Similarly 7, then 6, then 5 colours give *n*_{4} = 7, *n*_{5} = 6 and *n*_{6} = 5

Using the Product Rule: 6 x 5 x 8 x 7 x 6 x 5 = 50,400 possibilities. Notice that the first two elements were based on selection of position (as red and yellow are pre-selected), and the last four elements were based on selection of colour.

2. A random four-letter code is being generated from upper-case letters. A letter cannot be used more than once. The first letter may be any character; the remaining five letters must be consonants. How many combinations are there?

Answer: 183,540

There are two separate types of combination: vowel + three consonants; or four consonants.

Vowel+consonants: There are 5 vowels that could be in position 1; 21 consonants for position 2; 20 consonants for position 3 and 19 consonants for position 4 giving a total 5 x 21 x 20 x 19 = 39,900 possibilities.

Consonants only: for position 1: 21 possibilities, for position 2 there are 20, for position 3 there are 19 and for position 4 there are 18 = 21 x 20 x 19 x 18 = 143,640 possibilities.

The combined total is 39,900 + 143,640 = 183,540 possibilities.

Our iOS app has over 1,000 questions to help you practice this and many other topics.

Available to download free on the App Store.

Available to download free on the App Store.