A bag contains 4 red marbles 3 green marbles 5 yellow marble

A bag contains 4 red marbles, 3 green marbles, 5 yellow marbles and 2 blue marbles. Each marble is distinguishable. A) How many possible sets of 4 marbles are there? B) How many possible sets of 4 marbles contain no red marbles? C) How many possible sets of 4 marbles contain exactly one of each color?

Please show all work and explain your methods.

Solution

a) total marbles = 4+ 3+ 5+ 2 = 14

    possible sets of 4 marbles are = 14C4 = 1001

here any four marbles out of 14 are selected.

b)possible set of 4 marbles contains no red = 3C2 X 5C1 X 2C1 + 3C1 X 5C2 X 2C1 + 3C1 X 5C1 X 2C2

                                                               = 3 X 5 X 2 + 3 X 10 X 2 + 3 X 5 X 1

                                                               = 30 + 60 + 15

                                                               = 105

here if red is not to be selected, we have to select 2 marbles of same colour. here for 1st case, 2 green marbles are selected,for 2nd case, 2 yellow marbles selected and for third case 2 blue marbles are selected.

c) possible sets of 4 marbles contain exactly one of each color = 4C1 X 3C1 X 5C1 X 2C1 = 4 X 3 X 5 X 2 = 120

   here, one marble of each colour is selected.