Question

A bag contains 4 red marbles, 1 green one, 1 lavender one, 3 yellows, and 2 orange marbles. HINT [See Example 7.] How many sets of five marbles include at most one of the yellow ones?

Answer 1

266

Explanation:

Red marbles = 4

Green marbles = 1

Lavender marbles = 1

Yellow marbles = 3

Orange marbles = 2

To pick 5 marbles with at most 1 yellow, we can pick 5 non-yellow marbles or (4 non-yellow marble and 1 yellow marble).

Picking 5 non-yellow marbles

There are 4 + 1 + 1 + 2 = 8 non-yellow marbles.

The number of ways of picking any 5 is

`n_1 = \dbinom{8}{5} = 56`

Picking 4 non-yellow marble and 1 yellow marble

The number of ways of picking any 4 non-yellow marbles is

`n_2 = \dbinom{8}{4} = 70`

The number of ways of picking any 1 yellow marble from 3 is

`n_3 = \dbinom{3}{1} = 3`

Number of ways for both =
`70*3=210`

Total number of picking 5 marbles with at most 1 yellow

Therefore, total number of ways = 56 + 210 = 266