To determine how many different four-digit numbers Mike can make with the numbers 2, 2, 2, and 5, we need to calculate the number of distinct permutations of these four digits. Here's how we can do it step by step:
1. Identify the Total Number of Permutations:
First, calculate the total number of permutations of the four cards if all cards were distinct. The number of permutations of [tex]\( n \)[/tex] distinct objects is given by [tex]\( n! \)[/tex] (n factorial).
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
2. Adjust for Repeated Cards:
Next, adjust for the fact that some of the cards are not distinct. We have three '2' cards that are identical. To correct for these repeated elements, we divide the total number of permutations by the factorial of the number of repetitions. In this case, we need to divide by [tex]\( 3! \)[/tex] (the number of ways to arrange the three '2's among themselves).
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
3. Calculate the Number of Distinct Permutations:
Now, divide the total number of permutations by the number of permutations of the repeated cards:
[tex]\[
\frac{4!}{3!} = \frac{24}{6} = 4
\][/tex]
Therefore, Mike can make 4 different four-digit numbers using the cards 2, 2, 2, and 5.
So, the total number of different four-digit numbers he could make is [tex]\(\boxed{4}\)[/tex].
