Confused on permutation or combination on this question:
A VISA credit card has a first digit of 4 followed by 15 other digits. How many different VISA accounts does this allow?
I know the formulas but I am having a hard time determining which one to use. Help!!!
We have a choice of 10 digits at each of the following digits, so the total number of choices is
10^(15).
In real life, the last digit (16th) is used for validation purposes, and is not a free choice. There number of "free" digits is therefore limited to 14. But that's not part of the math question.
Thank you!
To solve this problem, we need to determine the number of different VISA accounts that can be created based on the given information.
Since the order of the digits matters (i.e., the first digit is specific), we should use the concept of permutations.
The formula for permutations is:
nPr = n! / (n - r)!
Where n is the total number of items and r is the number of items taken at a time.
In this case, we have 16 digits in total (1 digit for the first position and 15 digits for the remaining positions), and we want to find the number of different arrangements of these digits.
Substituting the values into the formula:
16P15 = 16! / (16 - 15)!
= 16! / 1!
Since 1! equals 1, we can simplify it further:
16P15 = 16!
Using a calculator or a factorial table, we can calculate 16!:
16! = 20922789888000
Therefore, the number of different VISA accounts that can be created is 20,922,789,888,000.
To solve this problem, we need to determine whether we should use the permutation or combination formula.
Permutations are used when the order of the elements matters, while combinations are used when the order does not matter. In this case, we are interested in the different VISA accounts, which are composed of 16 digits in total. However, the question does not specify whether the order of the digits matters or not.
Assuming that the order of the 15 other digits matters (meaning each digit can be different), we would use the permutation formula. The formula for permutations is:
P(n, r) = n! / (n - r)!
where n is the total number of possible elements and r is the number of elements taken at a time.
In this case, we have 16 digits in total, with the first digit fixed at 4. Hence, there are 15 other digits to choose from. To calculate the number of different VISA accounts, we can use the permutation formula:
P(15, 15) = 15! / (15 - 15)! = 15!
The factorial (symbolized by "!" after a number) denotes the product of all positive integers from 1 to that number. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.
Using a calculator or a computer program, we can calculate the value of 15! to be 1,307,674,368,000.
Therefore, there are 1,307,674,368,000 different VISA accounts if the order of the 15 other digits matters.
However, if the order does not matter (e.g., all 15 digits are the same), then we would use the combination formula instead. The formula for combinations is:
C(n, r) = n! / (r! * (n - r)!)
In this case, since the order of the digits does not matter, we would calculate the number of combinations of choosing 15 digits out of the possible options. Using the combination formula:
C(15, 15) = 15! / (15! * (15 - 15)!) = 15! / (15! * 0!) = 1
Therefore, if the order of the 15 other digits does not matter (i.e., they are all the same), there would be only 1 possible VISA account.
To summarize, if the order of the digits matters, use the permutation formula. If the order does not matter, use the combination formula.