SOLUTION: There are 20 different pairs of shoes - 8 black pairs, 6 red pairs, 3 white pairs, 2 yellow pairs, and 1 brown pair . How many selections of 3 pairs of shoes are possible if at lea

Algebra ->  Permutations -> SOLUTION: There are 20 different pairs of shoes - 8 black pairs, 6 red pairs, 3 white pairs, 2 yellow pairs, and 1 brown pair . How many selections of 3 pairs of shoes are possible if at lea      Log On


   

Question 1117931: There are 20 different pairs of shoes - 8 black pairs, 6 red pairs, 3 white pairs, 2 yellow pairs, and 1 brown pair . How many selections of 3 pairs of shoes are possible if at least 1 pair of shoes is black?
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

I will assume that the word  "different"  is applicable  UNIFORMLY  to all and each pairs/pair,  literally as the condition says it.

Thus each pair is considered as  DISTINGUISHABLE  and individual.

Then the way to solve the problem is  THIS: 

     First find the number of ALL POSSIBLE selections of 3 pairs 

     and then subtract all the selections NOT CONTAINING any black pair.


Solution 

1.  The number of all possible selections of 3 pairs (disregarding the color) is

        20*19*18 = 6840.



2.  Without 8 black pairs, there are 20-8 = 12 other pairs,

    and the number of selections of 3 pairs among these 12 pairs is

        12*11*10 = 1320.



3.  Thus the number under the question is the difference  6840 - 1320 = 5520.



Answer.  There are 5520 selections satisfying to the problem's requirements.