Question

(a) How many bit strings of length 8 are there?
(b) How many bit strings of length 8 or less are there?
(Count the empty string of length zero also.)
(c) How many strings of 6 lower case English letters are there that have the letter x in them somewhere? Here strings may use the same letter more than once. (Hint: It might be easier to first count the strings that don't have an x in them.)
(d) A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in 2 sizes for each sex. How many different types of this shirt are made?

Answer 1

(a) $2^8=256$

There are 256 bit strings of length 8.

(b)

$$2^0+2^1+2^2+2^3+2^4+2^5+2^6+2^7+2^8=$$

$$=2^9-1=511$$

There are 511 bit strings of length 8 or less.

(c) There are 26 letters in the alphabet.

The number of 6 letter strings $=26^6$

The number of 6 letter strings without any x $=25^6$

The number of 6 letter strings with at least one x

$$26^6-25^6=64775151$$

There are 64,775,151 strings of 6 lower case English letters that have the letter x in them somewhere.

(d)

Color $=12$

Gender $=2$

Size$=2$

Use the Product Rule

$$12\cdot2\cdot2=48$$

48 different types of this shirt are made.