thingy

There are always 3 ways to get to the next letter from any given letter: down-left, down, and down-right.

There are 3 transitions: A-R, R-C, and C-H

\(3\times3\times3=\boxed{27}\)

Q3

Obviously, the only way for that to happen is if they sit ACACAC.

We now permutate the children and permutate the adults.

\(3!3!=36\)

But, we need to correct for overcounting. The table has a 3-fold rotational symmetry and a 2-fold one. Therefore:

\(\frac{36}{6}=\boxed{6}\)

a)

There are 12 people and we want to pick 4 of them. So, we use

 \(\binom{12}{4} = 495\)

b)

First, we know that Mark has to be on the committee, so we put him in immediately. This leaves 3 spots to be filled by 11 people. Hence we use

\(\binom{11}{3} = 165\)

c)

We know that mark cannot be included, so there are 11 people who can go in 4 spots.

\(\binom{11}{4} = 330\)

I hope these are correct!