Conside all string whose letters belongs to the set Aabcd A

Conside all string whose letters belongs to the set A={a,b,c,d}

A) how many strings of length 6 are there that begin with a vowel and end with a vowel ?

B) how many strings of length 6 are there that contain exactly one a?

C) how many strings of length 6 are there that contain at least one a ?

D) how many strings of length 6 are there that contain at most one a?

E) how many strings of length 6 are there that contain exactly are vowel ?

F) how many strings of length 6 are there that contain exctly two consonants ?

g) how many strings of length 6 are there that contain extacly two or three consonants ?

H) how many strings of length 6 are there that contain at most on consonant ?

Solution

the only vowel in set A is a

a)

strings of length 6 that begin with vowel and end with vowel = 1 * 4 * 4 * 4 * 4 * 1 = 256

b)

strings of length 6 that contain exactly one a

= 4 * 3 * 3 * 3 * 3 * 3 = 972

c)

strings that contain atleast one a

= all strings - strings that doesnot contain a

= 4^6 - 3^6 = 3367

d)

strings that conatin atmost one a

= strings that doesnot contain a + strings that contain excatly one a

= 3367 + 972

= 4339

e)

strings that contain exactly one vowel = 972

f)

strings that contain exactly two consonents

= 3 * 3 * 1 * 1 * 1 * 1

= 9

g)

strings that contain 2 or 3 consonents

= 9 + 3 * 3 * 3 * 1 * 1 * 1

= 36

h)

strings that contain atmost one consonent = 4