To solve the problem of finding the total number of different 4-digit numbers that can be formed using all the digits in the number 4129, we follow these steps:
1. Identify the digits available: The digits we have are 4, 1, 2, and 9.
2. Determine the number of permutations: Since we want to use all the digits and each digit must appear exactly once in each 4-digit number, this is a problem of finding the permutations of the four digits.
3. Calculate the number of permutations: The number of permutations of a set of `n` distinct digits taken `n` at a time is given by `n!` (n factorial). In this case, `n` is 4 (since there are 4 digits: 4, 1, 2, and 9).
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
4. Conclusion: There are [tex]\(24\)[/tex] different 4-digit numbers that can be formed using all the digits in the number 4129. Moreover, since all digits are unique, every permutation of the digits results in a unique 4-digit number.
So, the total number of different 4-digit numbers that can be formed using the digits 4, 1, 2, and 9 is [tex]\(24\)[/tex].
