College Algebra & Trigonometry, 2018a

7.5. DISTINGUISHABLE PERMUTATIONS 327
7.5 Distinguishable Permutations
If there is a collection of 15 balls of various colors, then the number of permutations
in lining the balls up in a row is 15 P 15 = 15!. If all of the balls were the same
color there would only be one distinguishable permutation in lining them up in a
row because the balls themselves would look the same no matter how they were
arranged.
If 10 of the balls were yellow and the other 5 balls are all different colors, how
many distinguishable permutations would there be?
No matter how the balls are arranged, because the 10 yellow balls are indistinguishable
from each other, they could be interchanged without any perceptable
change in the overall arrangement. As a result, the number of distinguishable
permutations in this case would be 15! , since there are 10! rearrangements of the
10!
yellow balls for each fixed position of the other balls.
The general rule for this type of scenario is that, given n objects in which there are
n 1 objects of one kind that are indistinguishable, n 2 objects of another kind that
are indistinguishable and so on, then number of distinguishable permutations
will be:
Example
n!
n 1 ! ∗ n 2 ! ∗ n 3 ! ∗···∗n k !
with n 1 + n 2 + n 3 + ···+ n k = n
Find the number of ways of placing 12 balls in a row given that 5 are red, 3 are
green and 4 are yellow.
This would be 12!
5!3!4!
=
12 ∗ 11 ∗ 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1
5 ∗ 4 ∗ 3 ∗ 2 ∗ 3 ∗ 2 ∗ 4 ∗ 3 ∗ 2
=
12 ∗ 11 ∗ 10 ∗ 9 ∗ 8 ∗ 7 ∗ 6
3 ∗ 2 ∗ 4 ∗ 3 ∗ 2
=27, 720