How many seven-digit sequences of and are there that have a block of three successive but do not have a block of four or more successive ?

Source: mathcontest.olemiss.edu 2/17/2014

**SOLUTION**

Seven-digit binary sequences start from to , in decimal from to .

There are possible locations for a block of three successive zeros

1)

2)

3)

4)

5)

We examine each case in details.

1)

After putting a at the end of the block of three successive , , the three remaining bits can take on possible values from to . Case 1 has sequences that meet the requirement that they have a block of three successive but do not have a block of four or more successive .

2)

After sandwiching the block with two , , the two remaining bits can take on possible values from to . Case 2 has sequences.

3)

After sandwiching the block with two , , the two remaining bits can take on possible values from to . Case 3 has sequences.

4)

After sandwiching the block with two , , the two remaining bits can take on possible values from to . Case 4 has sequences.

5)

After putting a at the beginning of the block, , the three remaining bits can take on possible values from to . Case 5 would have sequences but for sequence which is a duplicate of Case 1.

Total =

**Answer**:

### Like this:

Like Loading...

*Related*

## About mvtrinh

Retired high school math teacher.