Suppose that a word is any string of seven letters of the al

Suppose that a \"word\" is any string of seven letters of the alphabet, with repeated letters allowed. (Recall that the alphabet has 26 letters, including 5 vowels.) How many words are there? How many words end with the letter T? How many words begin with a vowel or end with a vowel? How many words begin with AAB in some order? How many words have exactly one vowel?

Solution

(a) Each letter in the word can have 26 possible letters. So total words are:

26^7

(b)

Letter ends in T so last letter is fixed. First 6 letters have no restriction. So

26^6 words

(c)

Case 1: Words begin with a vowel

5 choices for first letter and 26^5 for remaining letters so:

5*26^5 words

Case 2: Words end with a vowel

5 choices for last letter. But now we need to be careful not to count case where words start with a vowel as that has been included in Case 1

So, 21 choices for first letter

So,

21*5*26^4=105*26^4

So total

5*26^5+105*26^4 words

e.

We can choose a position for the vowel in: C(6,1) ways and 5 choices for that letter.

Rest 5 letters have 21 choices each

So, C(6,1)*5*21^4=30*21^4