Range

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:
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: Thus, the number of arrangements is 109, which matches option (B).

Step 1: {Odd and even digits}
The digits of the number are . We have 4 odd digits and 5 even digits .
Step 2: {Placing odd digits in even positions}
We need to place the odd digits in the even positions of the nine-digit number. There are 4 even positions, and the number of ways to place the odd digits is:
Step 3: {Placing even digits in odd positions}
There are 5 odd positions remaining for the even digits. The number of ways to place the even digits is: Thus, the total number of ways is , which matches option (C).

Step 1: {Total number of people}
The total number of people is , and they are divided into two groups: one group of 4 persons and another group of 6 persons.
Step 2: {Number of ways to form groups}
The number of ways to form the two groups is .
Step 3: {Arrangements on the round tables}
For the group of 4 persons, the number of arrangements on a round table is . For the group of 6 persons, the number of arrangements on a round table is .
Step 4: {Total ways to arrange the guests}
The total number of ways to arrange the guests is: Thus, the total number of ways is , which matches option (B).