Mrs. wei was designing a 6 question true or false quiz. she did
not want there to be more than 2 answers the same in a row.
how many ways could the answer key be designed to fit the
criteria?

The answer key be designed to fit the  criteria is 12 ways for Mrs.Wei

Solution:

Given, Mrs. Wei was designing a 6 question True or False quiz.  

She did not want there to be more than 2 answers the same in a row.  

We have to find how many ways could the answer key be designed to fit the criteria?

Given that, more than 2 continuous answers are not allowed.

So, one answer same in a row conditions are t f t f t f or f t f t f t ⇒ 2 ways.

And, two answers same in a row conditions are  

Condition 1:

t t f f  t t  t t f t f t    t t f t t f

⇒3 ways by keeping 1st two answers same and constant and total 6 ways by interchanging true with false.

Condition 2:

f t t  f f t  f t t f t f  

⇒ 2 ways by keeping 2nd two answers same and constant and total 4 ways by interchanging true with false.

Condition 3:

Next, f f t t f f ⇒ this is covered in 1st case when we interchange true with false.so, we are done with possibilities.

So, total ways are 2 + 6 + 4 = 12 ways.

Hence, there are 12 ways.

a² + b²= c²

24² + 7² = c²

c= 25