Suppose we wanted to count the number of different possible bit strings of length five that start with either three 0’s or with two 1’s. In how many ways can we do that?

Question

discrete maths counting principle

Answer 1

Starting with 3 zeroes leaves 2 bits for which there are 4 (2²) combinations:

00000, 00001, 00010, 00011. These represent the decimal numbers 0-3.

Starting with 2 ones leaves 3 bits for which there are 8 (2³) combinations:

11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111. These represent the decimal numbers 24-31.

So there are 12 possible bit strings.