It is in fact the binomial distribution because the trials are independent of each other and you are counting the number of successes with a given number of trials.

For (a),

Let X be the number of heads.

P(X=12) = (20 C 12)*(0.5^12)*(0.5^8)

I'll let you do the computation yourself.

For (b),

Note from (a), the binomial distribution will always be a comination multiplied by 0.5^20

P(X>=12) = P(X=12)+P(X=13)+P(X=14)+...+P(X=20)

For each probability, just sub in the X value (eg. P(X=13) = (20 C X)*(0.5^20)

Once again, I'll leave the computation up to you.