Veronica finds eight sweaters she likes on sale in a store. How many different sweater purchases can she make
if money is no object?
Well, Veronica is really sweater-obsessed, isn't she? If money is no object, she can go ahead and buy all eight sweaters at once. That way, she can have a different sweater each day of the week! Talk about a fashion statement!
To calculate the number of different sweater purchases Veronica can make, we need to consider two factors: the number of sweaters she wants to buy and the number of sweaters available to choose from.
In this scenario, Veronica has eight sweaters that she likes and wants to buy. Since money is no object, she can buy any number of sweaters from this selection.
Now let's calculate the number of possible purchases she can make.
Since each sweater can be included or excluded from her purchase, we have two options for each sweater. Since Veronica has eight sweaters to choose from, the total number of different sweater purchases she can make can be calculated using the formula for combinations:
Total number of different purchases = 2^n
Here, n is the number of sweaters.
So, in this case, the number of different sweater purchases Veronica can make is 2^8 = 256.
If Veronica likes eight sweaters, she has the option to purchase any combination of those sweaters. To calculate the number of different sweater purchases she can make, we need to find the total number of possible combinations.
To find the total number of combinations, we can use the formula for combinations, which is expressed as:
C(n, r) = n! / (r! * (n - r)!)
where:
- n is the total number of objects (sweaters)
- r is the number of objects (sweaters) to be chosen at a time
In this case, Veronica has 8 sweaters to choose from, and she can choose any number of sweaters from 0 to 8 (inclusively). So, we need to calculate the sum of combinations for each possible value of r.
Let's calculate the number of different sweater purchases for each value of r:
- r = 0:
C(8, 0) = 8! / (0! * (8 - 0)!) = 1
- r = 1:
C(8, 1) = 8! / (1! * (8 - 1)!) = 8
- r = 2:
C(8, 2) = 8! / (2! * (8 - 2)!) = 28
- r = 3:
C(8, 3) = 8! / (3! * (8 - 3)!) = 56
- r = 4:
C(8, 4) = 8! / (4! * (8 - 4)!) = 70
- r = 5:
C(8, 5) = 8! / (5! * (8 - 5)!) = 56
- r = 6:
C(8, 6) = 8! / (6! * (8 - 6)!) = 28
- r = 7:
C(8, 7) = 8! / (7! * (8 - 7)!) = 8
- r = 8:
C(8, 8) = 8! / (8! * (8 - 8)!) = 1
Now, we add up all the different combinations for each value of r:
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256
Therefore, Veronica can make a total of 256 different sweater purchases if money is no object.