Question

Timothy is re-arranging his marble collection.he has five identical blue marbles,five identical green marbles and three identical black marbles he can fit exactly five marbles into a case and must have at least one of each.How many different ways can he arrange the case in?

Answer 1

720

Explanation:

Knowing that he must place one of each type, the total number of marbles must be one of theses 6 cases:

3 blues, 1 green, 1 black

2 blues, 2 green, 1 black

2 blues, 1 green, 2 black

1 blues, 2 green, 2 black

1 blues, 1 green, 3 black

1 blues, 3 green, 1 black

At the same time, each case can be ordered in many ways, for example in the first option:

blue, blue, blue, green, black

blue, blue, blue, black, green

blue, blue, black, blue, green

et cetera

The total number of permutations for each case is 5! = 120

Taking into account the 6 cases, 6 x 120 = 720

Answer 2

Based on the scenario, the case must be filled like this :

blue/green/black x blue/ green/black x blue/green/black x random x random

Different ways he can arrange it :

3 x 5!/(2!x2!) + 3x 5!/3!

90 + 60

= 150 ways

hope this helps