A math professor wears for each class a combination of : one of his three hats (red, blue, green); one of his pants (black, blue, white, green); one of his shirts (white, red, black); and one of his pair of shoes (black, red, blue, white). He never wears combinations with three items of the same color.

i. How many different combinations are wearable?

ii. The professor is teaching 70 lectures for the term. Can he wear a different combination for each lecture during one term?

iii. If he does wear a different combination each lecture, prove that he must use either the red or the blue shoes that term. v. There are 4 terms for the year. Prove that at least one combination will be used at least 3 times during the year.

iv. How many valid combinations have blue hat or shoes ?