Suppose a cabinet contains two blue cups, two red cups, two blue plates, and two red
plates. Suppose the four cups are placed uniformly at random on the four plates. Find the
probability that no cup is on a plate with a matching color.

Ans

Step-by-step explanation

Let: Cups are letters A, B, C, and D where: A and B = red cups C and D = blue cups Plates are letters E, F, G, and H where: E and F = blue plates G and H = red plates Suppose that there are only four ways to arranged the 2 blue cups, 2 red cups, 2 blue plates and 2 red plates successfully based on the given condition: ( A, E ), ( B, F ), ( C, G ), ( D, H ) ( A, E ), ( B, F ), ( C, D), ( D, G) ( A, F ), ( B, E ), ( C, G), ( D, H) ( A, F ), ( B, E ), ( C, H), ( D, G) Probability Analysis: To place the 2 blue cups, 2 red cups, 2 blue plates and 2 red plates based on the given condition, we can notice that there are 4 ways to place a cup on plate E, which primarily leaving 3 ways to place a cup on plate F, which then later on leaving 2 ways to place a cup on plate G, and then lastly, it will leaving 1 way to place a cup on plate H. Therefore, Number of ways = ( 4 ) ( 3) (2) (1) Number of ways = 24 ways Probability ( no cup is on a plate with a matching color ) = 4 / 24 Probability ( no cup is on a plate with a matching color ) = 1/ 6 Answer