Worked solution
Choose letters and then account for their order
A four-letter word can be formed from any of the 26 letters in the alphabet. Determine the number of such words with
(i) repetitions of the letters allowed,
(ii) different letters and containing exactly one vowel,
(iii) repetitions allowed and containing at least one vowel.
CJC · Discussion · Question 4
Answer
(i) 456976; (ii) 159600; (iii) 262495.
(i) Repetition allowed
Each of the four ordered positions has 26 choices: .
(ii) Exactly one vowel and no repetition
Using the source’s 5 vowels and 21 consonants, choose one vowel and three distinct consonants, then arrange the four different letters: .
(iii) At least one vowel
The complement consists of words containing only consonants, with possibilities. Subtract from the unrestricted total: .
Common mistakes
Here a word means an ordered letter string; it need not be a dictionary word.
Solution notes
All printed asks, working, counting factors and source annotations visually compared and retained. No marks are printed for this question.