How do you do this question?
A shipment of 25 television sets contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units.
To solve this question, we will use combinations.
(a) To find the number of ways to choose all good units, we need to select 4 units out of the 22 good units (since 3 units are defective).
The number of ways to select 4 units out of 22 can be calculated using the combination formula: nCr = n! / r!(n-r)!, where n is the total number of units and r is the number of units to be selected.
Using this formula, we have:
Number of ways to select all good units = 22C4 = 22! / (4!(22-4)!) = 7315
Therefore, there are 7315 ways to select four television units and receive all good units.
(b) To find the number of ways to choose two good units, we have:
Number of ways to select 2 good units = 22C2
Using the combination formula again:
Number of ways to select 2 good units = 22C2 = 22! / (2!(22-2)!) = 231
Therefore, there are 231 ways to select four television units and receive exactly two good units.
(c) To find the number of ways to select at least two good units, we need to consider two cases:
Case 1: Selecting all 4 good units
Number of ways to select all good units = 22C4 = 7315
Case 2: Selecting 3 good units and 1 defective unit
Number of ways to select 3 good units = 22C3
Number of ways to select 1 defective unit = 3C1
Using the combination formula:
Number of ways to select 3 good units and 1 defective unit = 22C3 * 3C1 = 1540
Total number of ways to select at least two good units = Number of ways in Case 1 + Number of ways in Case 2
= 7315 + 1540
= 8855
Therefore, there are 8855 ways to select four television units and receive at least two good units.
To solve this question, we will use the concept of combinations.
(a) To calculate the number of ways to purchase four good units, we need to select four units from the 22 good units in the shipment. Since order doesn't matter in this case, we can use combinations.
The formula for combinations is given by:
C(n, r) = n! / (r!(n - r)!)
Where n is the number of items to choose from, and r is the number of items to be chosen.
In this case, n = 22 (number of good units) and r = 4 (number of units to be chosen).
Using the formula, we can calculate the number of ways:
C(22, 4) = 22! / (4!(22 - 4)!)
= 22! / (4!18!)
= (22 * 21 * 20 * 19) / (4 * 3 * 2 * 1)
= 7315
Therefore, there are 7315 ways to purchase four good units.
(b) To calculate the number of ways to purchase two good units, we need to select two units from the 22 good units and two units from the three defective units.
Using the same formula, we can calculate the number of ways:
C(22, 2) * C(3, 2) = (22! / (2!(22 - 2)!) * (3! / (2!(3 - 2)!))
= (22 * 21 / (2 * 1)) * (3 * 2 / (2 * 1))
= 231
Therefore, there are 231 ways to purchase two good units.
(c) To calculate the number of ways to purchase at least two good units, we need to find the sum of the number of ways to purchase all good units and the number of ways to purchase two good units:
7315 + 231 = 7546
Therefore, there are 7546 ways to purchase at least two good units.
By applying the combination formula, we were able to calculate the number of ways to purchase a specific combination of good and defective units from the given shipment of television sets.
ways to get 4 good units: 1 (good, good, good, good)
ways to get two good:
GGBB
GBBG
GBGB
BBGG
BGBG
BGGB
ways to get at least two good:
above ways, plus
GGGG
PLus
GGGB
GBGG
GGBG
BGGG