Muhammad's strings

Probability Level 4

How many binary strings of length $10$ are there such that there are no consecutive zeros and an even number of ones?

This problem is shared by Muhammad A.

Details and assumptions

A binary string is a sequence of integers that are $1$ or $0$ . The length of a string refers to the number of integers in it.

As an explicit example, the binary string of length $3$ are $000, 001, 010, 011$ , $100, 101, 110, 111$ .

The answer is 72.

1 solution

Mei Li Staff
May 13, 2014

First, we observe that if there are $4$ or fewer ones, then two zeroes must be next to each other.

If there are $6$ ones, then the $4$ zeroes can go in $7$ possible positions, and there are $\binom{7}{4}$ ways to do this. If there are $8$ ones, then the remaining $2$ zeros can go in $9$ possible positions, giving $\binom{9}{2}$ ways. If there are $10$ ones, there are no zeros to be placed. Thus, the answer is $\binom{7}{4}+\binom{9}{2}+1=72.$

Muhammad's strings

The answer is 72.

1 solution

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