A boy has 4 red, 5 yellow and 5 green marbles. in how many ways can the boy arrange the marbles in a line if:

a. marbles of the same color are indistinguishable?

b. all marbles have different sizes?

(a) since the same colored marbles are indistinguishable, then Red1 in the first place and Red2 in the seond place is the same as Red2 in the first place and Red1 in the second place. (which means: do not duplicate the count)

Answer: 4! 5! 5! = 345,600

(b) since all of the marbles have different sizes, every combination counts.

Answer: 14! = 87, 178, 291, 200