there are 48 raincoats for sale at a local men clothing store. Twelve are black. If 6 raincoats are selected to be marked down, find the probability that exactly 3 will be black.

To find the probability that exactly 3 out of 6 selected raincoats will be black, we need to consider the total number of possible outcomes and the number of favorable outcomes.

1. Total number of possible outcomes:
The total number of ways we can select 6 raincoats from the 48 available is given by the combination formula (nCr), which calculates the number of ways to select r items from a set of n items. In this case, n = 48 and r = 6.
Therefore, the total number of possible outcomes is given by:
48C6 = (48!)/(6!(48-6)!) = 12,271,512.

2. Number of favorable outcomes:
We need to calculate the number of ways to choose exactly 3 black raincoats from the 12 available, multiplied by the number of ways to choose the remaining 3 raincoats from the 36 non-black raincoats.
This can be calculated as:
12C3 * 36C3 = (12!)/(3!(12-3)!) * (36!)/(3!(36-3)!)
= (12!)/(3!9!) * (36!)/(3!33!) = 2,776 * 7,976 = 22,164,176.

3. Probability calculation:
Finally, we can find the probability that exactly 3 out of 6 selected raincoats will be black by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Therefore, the probability is:
22,164,176 / 12,271,512 = 0.147 or 14.7%.

Hence, the probability that exactly 3 out of 6 selected raincoats will be black is 0.147 or 14.7%.

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