Question

A 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.

iv. There are 4 terms for the year. Prove that at least one combination will be used at least 3 times during the year.

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

Answer 1

The professon can have hats, pants, shirts and shoes in the folowing colors.

The total number of color combinations possible is

.

The possible combnations with at least 3 red is .

The possible combnations with at least 3 blue is .

The possible combnations with at least 3 green is 0.

The possible combnations with at least 3 black is .

The possible combnations with at least 3 white is .

.i) The different valid combinations that are wearable is .

ii) Since , he can wear a different combination for each lecture during one term.

iii) The number of valid combination with red shoes is,

The number of valid combination with blue shoes is,

​

The combinations avilable without either the red or the blue shoes is .

So he has to wear either red or blue shoes in that term.

iV) The number of lectures in i year is .

Therefore, at least one combination will be used at least 3 times during the year.

v) The valid combinations with blue hat is .

The valid combinations with blue shoes is

The number of valid combinations that have blue hat or shoes is .