Question

Problem 33.2ish. How many strings of fourteen lowercase English letters are there which

(a) start with the letter x, if letters may be repeated?

(b) contain the letter x at least once, if letters can be repeated?

(c) contain each of the letters x and y at least once, if letters can be repeated?

(d) which contain at least one vowel, where letters may not be repeated?

Answer 1

Number of options for a letter, if there is no condition = 26

a) Number of strings that start with x, if letters may be repeated = 1 x 26¹³

= 2.481x10¹⁸

b) Number of strings that contain letter x at least once, if letters may be repeated = Total number of strings possible - Number of strings that does not contain the letter x

= 26¹⁴ - 25¹⁴

= 2.726x10¹⁹

c) Number of strings that contain letter x and y at least once, if letters may be repeated  = Total number of strings possible - Number of strings that does not contain the letter x - Number of strings that does not contain the letter y + Number of strings that does not contain the letters x and y

= 26¹⁴ - 25¹⁴ - 25¹⁴ + 24¹⁴

= 1.104x10¹⁹

d) Number of vowels = 5

Number of consonants = 21

If letters cannot be repeated, use permutation.

Permutation formula: nPr = n!/(n-r)!

Number of strings with at least one vowel = Number of strings possible - Number of strings in which there are no vowels

= 26P14 - 21P14

= 8.318x10¹⁷