a shipment of 10 microwaves contains 3 defective units. in how many ways can a vending company choose 4 units if: a)all units are good b)2 units are good c) at least 2 units are good
To answer these questions, we can use combinations and the concept of counting.
a) To find the number of ways the vending company can choose 4 units if all units are good, we need to calculate the combination of 10 choose 4. Since all units are good, we don't need to consider the defective units. The formula for combinations is:
nCr = n! / (r!(n-r)!)
Applying this formula:
10C4 = 10! / (4!(10-4)!) = 10! / (4!6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
Therefore, the vending company can choose 4 units from 10 good units in 210 different ways.
b) To find the number of ways the vending company can choose 4 units if 2 units are good, we need to choose 2 good units out of 10 and 2 units out of 3 defective ones. Again, we can use combinations.
The number of ways to select 2 good units from 10 is given by:
10C2 = 10! / (2!(10-2)!) = (10 * 9) / (2 * 1) = 45
Similarly, the number of ways to select 2 defective units from 3 is:
3C2 = 3! / (2!(3-2)!) = (3 * 2) / (2 * 1) = 3
Now, to find the total number of ways to select 4 units with 2 good units, we multiply the number of ways to select 2 good units and 2 defective units:
45 * 3 = 135
Therefore, the vending company can choose 4 units with 2 good units in 135 different ways.
c) To find the number of ways the vending company can choose 4 units if at least 2 units are good, we need to add up the possibilities where exactly 2, exactly 3, or all 4 units are good.
The number of ways to select exactly 2 good units and 2 defective units can be calculated using the same method as in part b, which is 135.
The number of ways to select exactly 3 good units and 1 defective unit is:
10C3 = 10! / (3!(10-3)!) = (10 * 9 * 8) / (3 * 2 * 1) = 120
Lastly, the number of ways to select all 4 good units is:
10C4 = 210
We add up these three possibilities:
135 + 120 + 210 = 465
Therefore, the vending company can choose 4 units with at least 2 good units in 465 different ways.