Monika removed the number $123.412$ from the board, which was assembled using magnetic digits. She wants to use these digits to create a 4-digit number. How many different 4-digit numbers can she form from the given magnetic digits?
Monika noticed that some digits are repeated. She doesn't like that, as it makes the task more complicated. However, she decided to tackle the problem!
(1) First, she calculated the number of 4-digit numbers in which no digits are repeated. There are $4! = 24$ such numbers.
(2) Next, she calculated the count of numbers with two ones:
- She can place two ones in a 4-digit number in six ways ($11xy$, $1x1y$, ...).
- Each placement of ones can be complemented by the remaining distinct digits in $P(3,2) = 3! = 6$ ways.
- The total number of 4-digit numbers that will contain exactly $2$ ones is $6 \cdot 6 = 36$.
(3) She concluded that the count of numbers containing exactly two twos will be the same as the count of numbers containing exactly two ones, which is $36$.
(4) Then, Monika considered numbers containing two ones and two twos. According to Monika, there are $12$ such numbers: $\frac{4!}{2!}=12$.
(5) In conclusion, Monika stated that she can create 4-digit numbers using the magnetic digits only in the ways described in steps (1) through (4). Therefore, she can assemble $24 + 36 + 36 + 12 = 108$ different numbers using the magnetic digits.
Did Monika make a mistake in her calculation? If yes, determine, in which step.
Monika managed to solve the problem flawlessly despite her concerns!
Monika made her first mistake in step (1). The number of 4-digit numbers in which no digits are repeated is actually $2 \cdot 4! = 48$. So, the total number of different numbers that can be formed from the magnetic digits is $48 + 36 + 36 + 12 = 132$.
Monika made her first mistake in step (2). Two ones can be arranged in a 4-digit number in six different ways, but we can complement them with the remaining digits in $P(4,2) = 12$ ways. Hence, the number of 4-digit numbers that will contain exactly $2$ ones is $6 \cdot 12 = 72$. Therefore, the total number of different numbers that can be formed is $24 + 72 + 72 + 12 = 180$.
Monika made her first mistake in step (4). The number of numbers containing two ones and two twos is actually half of what she calculated, i.e., $\frac{4!}{2! \cdot 2!} = 6$. Consequently, the total number of different numbers that can be formed is $24 + 36 + 36 + 6 = 102$.