Solution: 2017 Fall Midterm - 10

Author: Michiel Smid

Question

A bitstring is called 01-free if it does not contain 01. Let $n \geq 2$ be an integer. How many 01-free bitstrings of length $n$ are there?

(a)

$n-1$

(b)

$n+1$

(c)

$n+2$

(d)

$n$

Let’s use the example where $ n = 4 $

The first possibility is that it starts with 0 and the rest of the bits are 0

0000

This is 1 possibility.

The second possibility is that it starts with a row of 1’s and the rest of the bits are 0

1111

1110

1100

1000

This is 4 possibilities.

So we have $ 1 + n $ possibilities.

For short bitstrings with simple equations, you can also brute force the possibilities.

Suppose $ n=2 $

It can be 00, 10, 11

There are 3 possibilities.

Since it’s more than n by 1, we can say that there are $ n+1 $ possibilities.