finite Math Question

Answer:
12!/(3! * 4! * 5!)
= 479001600/(6 * 24 * 120)
= 27720

Explanation 1:
Out of 12 officers, choose 3 from them to be assigned to Task 1.
The number is 12C3.
In the remaining 12 - 3 = 9 officers, choose 4 from them to be assigned to Task 2.
The number is 9C4.
For the final remaining 9 - 4 = 5 officers, choose all 5 to be assigned to Task 3.
The number is 5C5.
The total number of ways is
12C3 * 9C4 * 5C4
= 12!/(3! * 9!) * 9!/(4! * 5!) * 5!/(5! * 0!)
= 12!/(3! * 4! * 5!)

Explanation 2:
Consider the permutation of 12 officers in a row.
There are 12! of ways.
Assign the first 3 to do Task 1, the next 4 to do Task 2, the last 5 to do Task 3.
Then, the arrangements within the first 3 do not affect the overall assignment, thus the above number needs to be divided by 3!.
Similarly, for the middle 4 and the final 5, the total number needs to be divided by 4! and 5! for the two groups.
Thus, the total number of ways is
= 12!/(3! * 4! * 5!)