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 out how many different four-digit numbers can be obtained using the digits 5, 1, 1, 5, we can break it down into smaller steps:
Step 1: Determine the total number of arrangements of all four digits.
Since we have four digits, we can arrange them in 4! (4 factorial) ways. The factorial of a number means multiplying the number by all positive integers less than it. Therefore, 4! = 4 x 3 x 2 x 1 = 24.
Step 2: Calculate the number of duplicate arrangements.
In this case, we have duplicates of the digit 1, so we need to divide the total number of arrangements by the number of duplicate arrangements. Since we have two identical digits "1," we need to divide the result from Step 1 by 2! (2 factorial). 2! = 2 x 1 = 2.
Step 3: Calculate the final number of distinct four-digit numbers.
To obtain the number of distinct four-digit numbers, we divide the result from Step 2 by 2! because we also have two identical digits "5." Hence, the number of different four-digit numbers that can be obtained by using all four digits 5, 1, 1, 5 is:
24 / (2! x 2!) = 24 / (2 x 2) = 24 / 4 = 6
Therefore, the answer is 6 different four-digit numbers can be obtained using the digits 5, 1, 1, 5.