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

For the number 12345, determine the number of digit arrangements where a minimum of 3 digits are displaced from their original positions. This is a derangement problem focused on counting digits not in their initial locations.
Step 1: Compute the total permutations of the 5 digits.
The total number of permutations for 5 digits is calculated as: \[ 5! = 120. \] 
Step 2: Compute derangements for scenarios involving 3, 4, or all 5 digits out of place.
Applying the principle of inclusion-exclusion, the count of arrangements with at least 3 digits not in their original positions is: \[ 5C3 \times 3! - 5C4 \times 4! + 5C5 \times 5! = 20 + 45 + 44 = 109. \] Consequently, there are 109 such arrangements, corresponding to option (B).