SOLUTION: A bag contains 4 red marbles, 3 green ones, 1 lavender one, 3 yellows, and 2 orange marbles. How many sets of five marbles include at least three red ones?

Question 1161125: A bag contains 4 red marbles, 3 green ones, 1 lavender one, 3 yellows, and 2 orange marbles.
How many sets of five marbles include at least three red ones?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

"At least 3" means "3 or more". Considering there are 4 red marbles, the only possible scenarios are:

picking exactly 3 red marbles
picking exactly 4 red marbles

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Let's consider the situation where we have exactly 3 red marbles selected. There are 4 red marbles overall, which means there are 4 ways to form a group of 3 selected red. Put another way, there are 4 ways to not a select a particular red marble.

Using the combination function, you should find that

when n = 4 and r = 3.

There are 3 green marbles, 1 lavender, 3 yellows and 2 orange giving a total of 3+1+3+2 = 9 marbles that aren't red. Use the combination function to find

when n = 9 and r = 2. This says there are 36 ways to select two marbles from a pool of 9.

Multiply this with the number of ways to pick the red marbles (4) and we get 36*4 = 144

There are 144 ways to pick exactly 3 red marbles and 2 non-red marbles.

Let A = 144. We'll use it later.

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We have 4 red marbles. There is only one way to select a group of 4 red marbles if the pool size is also 4.

There are 9 non-red marbles and only one slot to fill with this pool to pick from. So we have 9 different ways to fill it.

Overall, there are 1*9 = 9 different ways to pick four red marbles followed by one non-red marble.

Let B = 9

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The results from the earlier sections are then added up. The values of A and B can be added because the events they represent are mutually exclusive.

A+B = 144+9 = 153

Answer: 153

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