Indicate whether the following situation involves a combination
or a permutation. Explain.
An 8-digit code chosen for a lock
This situation involves a permutation because the order of the digits in the code matters. Changing the order of the digits would result in a different code and potentially allow access to the lock.
Well, selecting an 8-digit code for a lock involves a combination. A combination refers to the arrangement of items where the order doesn't matter. In this case, the order of the digits in the code doesn't matter as long as the correct digits are used. So, it's a combination since you can't go around messing up the order of the code and expect the lock to open. Trust me, I tried that once, and the lock didn't find it amusing at all.
The situation of choosing an 8-digit code for a lock involves a permutation.
A permutation is an arrangement of objects in a specific order without repetition, while a combination is a selection of objects without regard to their order. For example, if we were selecting a 4-digit code where repetition was allowed, then it would be a combination. But in this case, choosing an 8-digit code for a lock means that we cannot repeat any digits.
In a permutation, the order of the elements matters. Each digit in the code can be different, and the arrangement of these digits will result in a different code. Therefore, selecting an 8-digit code for a lock involves choosing and arranging 8 digits in a specific order, making it a permutation.
The situation of selecting an 8-digit code for a lock involves a permutation.
A permutation is an arrangement or ordering of a set of objects, where the order matters. In this case, the digits of the code have a specific order and changing the order would result in a different code. Each digit can be different from the others, and repetition is not allowed since it would create different codes.
To calculate the number of possible permutations for the 8-digit code, we can use the formula for permutations of n objects taken r at a time:
P(n, r) = n! / (n - r)!
In this situation, we have 10 digits (0 to 9) to choose from for each position of the 8-digit code. So, we have:
n = 10 (number of options for each digit)
r = 8 (number of digits in the code)
Plugging these values into the formula:
P(10, 8) = 10! / (10 - 8)!
= 10! / 2!
Simplifying:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
2! = 2 * 1
P(10, 8) = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1)
= 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3
= 40,320
Therefore, there are 40,320 possible permutations for the 8-digit code chosen for the lock.