Question

The number of arrangements of all digits of 12345 such that at least 3 digits will not come in its position is:

• A. 89
• B. 109
• C. 78
• D. 57

Hint:

Derangements are used to count the number of permutations where no element appears in its original position. Use inclusion-exclusion for such problems.

Answer 1

The Correct Option is B

We are given the number 12345, and we need to find how many arrangements of its digits will result in at least 3 digits not being in their original position. This is a problem of derangements where we calculate how many digits do not appear in their original position. 
Step 1: Calculate the total number of arrangements of 5 digits
The total number of arrangements of 5 digits is: \[ 5! = 120. \] 
Step 2: Calculate derangements for different cases (3 digits, 4 digits, and 5 digits out of position)
Using inclusion-exclusion, the number of ways that at least 3 digits do not appear in their original position is: \[ 5C3 \times 3! - 5C4 \times 4! + 5C5 \times 5! = 20 + 45 + 44 = 109. \] Thus, the number of arrangements is 109, which matches option (B).