Example 20
Seven cards, each bearing a letter, can be arranged to spell the word DOUBLES. A code word
is formed by arranging three cards randomly. Find, in form, the probability that the code word
(a) contains the letter S,
(b) consists of a vowel between two consonants?

To find the probability that the code word contains the letter S, first determine the total number of possible arrangements of three letters from the seven available letters (D, O, U, B, L, E, S). The total arrangements are given by . Next, we find the number of arrangements containing at least one S. The scenarios include S being in the first, second, or third position, leading to a total of . Thus, the probability that the code word contains S is .

For the second part regarding the arrangement consisting of a vowel between two consonants, we first identify the vowels (O, U, E) and consonants (D, B, L, S). To form the arrangement, we can choose one vowel (3 options) and two consonants (from D, B, L, S), which can be selected in ways. The arrangement of the chosen letters in the format CVC (consonant-vowel-consonant) gives us valid code words. Therefore, the probability for this scenario is .