The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ______.
The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to ______.
Solution and Explanation
The number of four-digit numbers that can be formed using the digits 2, 1, 2, and 3 is \( \frac{4!}{2!} = 12 \).
These are the permutations of the digits 2, 1, 2, and 3. The sum of digits at the unit place is calculated as: \[ 3 \times 1 + 6 \times 2 + 3 \times 3 = 24. \] Now, the required sum is: \[ 24 \times 1000 + 24 \times 100 + 24 \times 10 + 24 \times 1 = 24 \times (1000 + 100 + 10 + 1) = 24 \times 1111 = 26664. \] Thus, the sum is \( 26664 \).
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Concepts Used:
Permutations
A permutation is an arrangement of multiple objects in a particular order taken a few or all at a time. The formula for permutation is as follows:
\(^nP_r = \frac{n!}{(n-r)!}\)
nPr = permutation
n = total number of objects
r = number of objects selected
Types of Permutation
- Permutation of n different things where repeating is not allowed
- Permutation of n different things where repeating is allowed
- Permutation of similar kinds or duplicate objects