TITLE: Why is the following method incorrect when deciding on the possible choices?
QUESTION [7 upvotes]: "Six men and three women are standing in a supermarket queue. Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?"
I went about solving this question by considering 3 cases, first with a single woman and then with 2 and finally with 3.
$$
 {3 \choose 1} \ \times \ {6 \choose 2} \ + \  {3 \choose 2} \ \times \ {6 \choose 1}  \ + \  {3 \choose 3}
$$
Which simplifies to $ \ 45 \ + \ 18 \ + \ 1$ leading to $64$ different choices.
This approach is in fact correct. However, another approach came to my mind as well.
If at least 1 woman must be included in the group then we can simply choose 1 from the 3 women, and fill the remaining 2 slots from the remnant 8 'people'.
$$
{3 \choose 1} \ \times \ {8 \choose 2}
$$
This, however, gives the answer $84$. This answer is most certainly wrong but I am unable to explain why the method is incorrect. If someone could explain why this leads to the wrong answer that would be very nice.

REPLY [5 votes]: There is a much simpler approach. "At least one woman" is the opposite of "zero women" or "all men" from the total, so subtracting from the total gives us:
$${9 \choose 3} - {6 \choose 3} = 64$$

If you are interested in another subtractive approach, following from N. F. Taussig's answer, you can again subtract from ${3 \choose 1} \cdot {8 \choose 2} = 84$.
The cases where there are two women and one man (Abigail, Beatrice, Charles) are counted $2$ times, when they should only be counted $1$ time. Therefore, you need to subtract the number of combinations of this type $1$ time. This is just ${3 \choose 2} \cdot 6 = 18$.
Similarly, the cases where there are three women are counted $3$ times, when they should only be counted $1$ time, as there is only one combination possible with all $3$ women. Therefore, the number of combinations should be subtracted by $2$.
Hence $84 - 18 - 2 = 64$, which is the correct answer.