TITLE: Permutation and Combination: Why is my logic incorrect?
QUESTION [5 upvotes]: So, the question is, 
"In how many ways can 6 persons be selected from 4 officers and 8 constables if at least one officer is to be included?"
I did this,
4C1 x 11C5
I first selected one officer from the 4 officers and then selected remaining 5 from the 11 persons remaining. 
The answer I'm getting is wrong, the correct one is 896 which I got by subtracting the total number of combinations by using only constables minus all the combinations, 
12C6 - 8C6
It would be great if someone can point out the mistake I committed in my first logic.
Thanks!

REPLY [10 votes]: Suppose the officers are ABCD and the constables are 01234567.
What you are counting are pictures of the shape
{ one letter } | { five characters }

Here are three different pictures of this shape
A | BC567
B | AC567
C | AB567

However, all three of these pictures describe the same selection ABC567. Thus, you've counted this single selection three different times, and get the wrong answer.

REPLY [4 votes]: Well, you pick the first person name $A$ and choose 5 remains which contains an officer name $B$.
After that, you pick the first person as $B$ and in 5 people are the same with the first way, but change $A$ to $B$. Two ways are the same, then you count twice.