(a) Suppose each single character stored in a computer uses eight bits. Then cach character is represented by a different sequence of eight 0's and l's called a bit pattern. How many different bit patterns are there? (That is, how many different characters could be represented?) (b) How many bit patterns are palindromes (the same backwards as forwards)? (c) How many different bit patterns have an even number of 1's?

Step 1
a) is relatively straight forward. For the first bit, there are 2 possibilities: 1 and 0. For the first two, the possibilities are 11, 10, 01, and 00. 4 in total. You will find that as you continue this trend, the number of combinations equals $2^n$ where n is the length of the character.
Step 2
b) is similar. A palindrome is the same forwards as backwards. Therefore the last 4 numbers equal the first 4 in reverse order. Now there are only 4 digits to work with. Use the same system in a) as here, just with 4 instead of 8.
Step 3
c) is a bit tricky. For that one, you simply reduce the power by 1. So with an 8 bit binary string with only even amounts of 1s, you would result with $2^{n-1}$ possibilities. In this case, $2^7$.