A store clerk filing an order for 17 models of a new smart-phone inadvertently pulled some older models and included them in the order. There are 25 new models and 12 older models in stock when the clerk filled the order. Determine the probability that the company receiving the order has 6 or more old models in their order.
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To determine the probability that the company receiving the order has 6 or more old models, we need to use the concept of combinations and probability.
First, let's calculate the total number of ways the clerk can choose 17 models from the total stock of 37 models (25 new + 12 old). This can be done using the combination formula:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of items and k is the number of items we want to choose. In this case, n = 37 and k = 17.
C(37, 17) = 37! / (17!(37-17)!)
= 37! / (17!(20!))
= (37 * 36 * 35 * ... * 21) / (17 * 16 * 15 * ... * 1 * 20 * 19 * ... * 1)
Now, let's calculate the number of ways the clerk can choose 17 models, but with 6 or more old models. We need to consider two cases: when the clerk chooses exactly 6 old models, and when the clerk chooses 7, 8, 9, ... , or 12 old models.
Case 1:
Choosing exactly 6 old models:
The clerk needs to choose 6 old models from the available 12 old models, and choose 11 new models from the available 25 new models. We can calculate this using the combination formula.
C(12, 6) = 12! / (6!(12-6)!)
= 12! / (6!(6!))
= (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1)
Choosing the remaining 11 models from the new models:
C(25, 11) = 25! / (11!(25-11)!)
= 25! / (11!(14!))
= (25 * 24 * 23 * ... * 15) / (11 * 10 * 9 * ... * 1 * 14 * 13 * ... * 1)
Now, we calculate the total number of ways to choose 17 models with exactly 6 old models by multiplying the two values:
Total ways = C(12, 6) * C(25, 11)
Case 2:
We repeat the same calculation for choosing 7, 8, 9, 10, 11, and 12 old models, and calculate the total number of ways for each case.
Finally, we add up the total number of ways for each case:
Total ways = (total ways for 6 old models) + (total ways for 7 old models) + ... + (total ways for 12 old models)
Now, to find the probability, we divide the total ways with 17 models having 6 or more old models by the total number of ways to choose 17 models from the total stock.
Probability = Total ways / C(37, 17)
Now, you can plug in the values and calculate the exact probability.