how many different four digit numbers can be obtained by using all four of these digits 5,1,1,5? show work please

To find how many different four-digit numbers can be obtained by using all four digits (5, 1, 1, 5), we can use the concept of permutation.

A permutation is an arrangement of objects in a specific order. In this case, we want to find the number of different four-digit numbers, so we need to find the number of permutations of the given digits.

To calculate the number of permutations, we use the formula:

nPr = n! / (n-r)!

Where n is the total number of objects (in this case, 4 digits) and r is the number of objects to be arranged (in this case, 4 digits).

Let's calculate the number of permutations step by step:

1. Count the frequency of each digit: 5 appears twice, and 1 appears twice.

2. Calculate the total number of objects (n):
n = 4 (total number of digits)

3. Calculate the number of arrangements required (r):
r = 4 (total number of digits to be arranged)

4. Apply the permutation formula:
nPr = n! / (n-r)!
= 4! / (4-4)!
= 4! / 0!
= 4! / 1
= 4 x 3 x 2 x 1 / 1
= 24

Therefore, the number of different four-digit numbers that can be obtained by using all four digits (5, 1, 1, 5) is 24.