Answer
a) 26x^{8}
b)26 x 25 x 24 x 23 x 22 x 21 x 20 x 19
c) 1 x 26^7
d) 1 x 26 x 25 x 24 x 23 x 22 x 21 x 20
e) 1 x 26^6 x 1
f) 1 x 1 x 26^6
g) 1 x 1 x 26^4 x 1 x 1
h) 26^6 + 26^6 - 26^4
Work Step by Step
a) 26x^{8}
There are 26 letters in the alphabet thus there are 26 options for each position in the 8-length string
b)26 x 25 x 24 x 23 x 22 x 21 x 20 x 19
There are 26 letters in the alphabet thus there are 26-n1 options for each position in the 8-length string, where n is a the position index.
c) 1 x 26^7
first position has 1 option which is X, while others can be any letter from the 26 letters
d) 1 x 26 x 25 x 24 x 23 x 22 x 21 x 20
first position has 1 option which is X, while others can be any letter from the 26 letters such that each letter, where every time we pick a letter we decrement the letter option by 1.
e) 1 x 26^6 x 1
First position is for the letter X, other 6 position are for any letter in the alphabet, the last position is for X again.
f) 1 x 1 x 26^6
First two positions are for BO, others can be any letter in the alphabet.
g) 1 x 1 x 26^4 x 1 x 1
First two positions are for BO, other 4 can be any letter in the alphabet, last two positions are for BO again
h) 26^6 + 26^6 - 26^4
Number of strings that start with BO + Number of strings that end with BO - Number of strings that start and end with BO
