9 pieces of fruit are slected at random from a bowl containing 4 oranges, 6 nectarines and 5 apples. Calculate the probability that there are 3 of each type of fruit in the selection.
To calculate the probability that there are 3 of each type of fruit in the selection, we first need to determine the total number of possible selections of 9 fruits from the bowl.
The total number of fruits in the bowl is 4 oranges + 6 nectarines + 5 apples = 15 fruits.
Now, we can use combination formula to calculate the total number of possible selections. The combination formula is:
nCr = n! / (r! * (n-r)!)
where n is the total number of items and r is the number of items being selected.
In this case, we have 15 fruits (total) and we want to select 9 fruits. Therefore, the total number of possible selections is:
15C9 = 15! / (9! * (15-9)!)
= 15! / (9! * 6!)
= (15 * 14 * 13 * 12 * 11 * 10)/(6 * 5 * 4 * 3 * 2 * 1)
= 5005
So, there are 5005 possible selections of 9 fruits from the bowl.
Next, we need to determine the number of favorable or successful outcomes, which in this case is the number of ways to select exactly 3 oranges, 3 nectarines, and 3 apples.
The number of ways to select 3 oranges from the 4 available is:
4C3 = 4! / (3! * (4-3)!)
= 4! / (3! * 1!)
= 4
Similarly, the number of ways to select 3 nectarines from the 6 available is:
6C3 = 6! / (3! * (6-3)!)
= 6! / (3! * 3!)
= 20
Finally, the number of ways to select 3 apples from the 5 available is:
5C3 = 5! / (3! * (5-3)!)
= 5! / (3! * 2!)
= 10
To find the total number of favorable outcomes, we multiply the number of ways to select 3 oranges, 3 nectarines, and 3 apples:
Total favorable outcomes = 4 * 20 * 10 = 800
Now, we can calculate the probability by dividing the total number of favorable outcomes by the total number of possible selections:
Probability = Total favorable outcomes / Total possible selections
= 800 / 5005
≈ 0.1597
Therefore, the probability that there are 3 of each type of fruit in the selection is approximately 0.1597 or 15.97%.