Rachel has 14 red, 4 blue, and 3 yellow towels. In how many ways can the towels be arranged if one blue towel is at each end of the row?

Question

Rachel has 14 red, 4 blue, and 3 yellow towels. In how many ways can the towels be arranged if one blue towel is at each end of the row?

Answer 1

14 red, 4 blue, and 3 yellow

1 blue on each end means we only have 2 blue in our arrangement in between the blue ones

14 red, 2 blue, and 3 yellow

how many ways can we arrange them ?

14+2+3=19 towels to be arranged

how many ways can we arrange the 19 towels keeping in mind there are repetitions (14 of the same color, 2 of the same color, and 3 of the same color) ?

(19 !)/[(14 !)(2 !)(3 !)]

(19*18*17*16*15)/(2*1)(3*2*1)

the 14*13*12*...3*2*1 in the numerator cancels with the 14 ! in the denominator

1,395,360/12=116,280 arrangements.