There is this problem I found in a book on combinations. It goes roughly like this:
There are 4 type A machines and 5 type B machines. Three of the machines are removed from the piles. What is the total number of possibilities that the result contains at least one A machine and at least one B machine?
The answer on the book was given as 1C4 * 2C5 + 2C4 * 1C5 = 70 total possibilities. However, the way I did it was by first enumerating the total methods of choosing one type A and one type B, i.e. 20 different methods, and then multiplying by 7, the number of machines left over after choosing 2. This results in an answer of 140, two times too much.
Why was my answer wrong?