Explain how you would calculate the total number of distinct 5-digit PIN codes possible when each digit can range from 0 to 9, with the stipulation that no digit is repeated.
user
Sure, I can explain how to calculate the total number of distinct 5-digit PIN codes possible with the given conditions.

As each digit can range from 0 to 9, we have a total of 10 choices for the first digit. But in a PIN, we don't want any two digits to be the same, so we eliminate one option: 12345. Therefore, 10 choices becomes 9 choices.

Now, for the second digit, the available options are from 0 to 9, except for what the first digit can't be. That is, for each position, the available digits are the ones that have not yet been selected. So, for the second digit, we have 8 choices (since the digit 00xx is already taken).

Next, for the third digit, there are 7 choices, then 6, and so on.

So, for calculating the total number of possibilities, we will use a combinatorics formula.

The formula for combination is (n choose r), which is given by n! / (r! * (n-r)!). Here, n is the total number of items, and r is the number of items selected in each combination.

In this case, n = 10, and r = 5 (digit 00000 is already taken).

Therefore, the total number of feasible combinations is,

Total combinations = (10 choose 5) = (10 * 9 * 8 * 7 * 6) / (5 * 4 * 3 * 2 * 1) = 252

Therefore, the possible numbers of PIN combinations without any repeated digits are 252.
assistant
Explain how you would calculate the total number of distinct 5-digit PIN codes possible when each digit can range from 0 to 9, with the stipulation that no digit is repeated.
user
To calculate the total number of distinct 5-digit PIN codes possible when each digit can range from 0 to 9, with the stipulation that no digit is repeated, we'll use the concept of permutations.

A permutation is an arrangement of objects without repetition where the order matters. The formula for permutations is:

P(n, r) = n! / (n - r)!

Where "n" represents the total number of items to choose from, "r" represents the number of items to choose, and "!" denotes the factorial of a number (the product of all positive integers up to that number).

However, in this case, we cannot use the formula directly because we cannot have a leading zero in our 5-digit PIN. So, we'll break down the calculation into two steps:

1. Calculate the number of 5-digit PINs starting with a non-zero digit.
2. Calculate the total number of 5-digit PINs.

Step 1: For 5-digit PINs starting with a non-zero digit:
We have 9 choices for the first digit (any digit except 0), 9 choices for the second digit (any remaining digit), 8 choices for the third digit, 7 choices for the fourth digit, and 6 choices for the fifth digit.

So, the number of 5-digit PINs starting with a non-zero digit is:
9 * 9 * 8 * 7 * 6 = 27,9936

Step 2: Total number of 5-digit PINs:
Since all 5-digit PINs must start with a non-zero digit, the total number of distinct 5-digit PIN codes possible with no repeated digits is equal to the number of PINs starting with a non-zero digit.

Total 5-digit PINs = 27,9936

Thus, there are 27,9936 distinct 5-digit PIN codes possible with no digit repetition when considering the range 0-9.
assistant