How to figure combinations?

Question

I know how to get the total combinations in a group: e.g. how many combinations of three in a group of 12 = 12!/(12-3)! But I don't know how to factor in limitations.

12 people, of whom two are teachers. How many groups of three can you have that have at least one teacher?

Can you give me a formula and/or explain it so I can make logical sense of it? Thx...

Answer 1

see what u do is first read the question once again. what is given is u have to make a group of 3 persons consisting of atleast 1 teacher and u have max 2 teachers so what u do is the foll

10C2 * 2C1  +  10C1 * 2C2  =  100

see out of 12 2 are teachers so separate them n u r left with 10 ppl.

now u choose 1 teacher from 2 teachers and 2 ppl from 10 ppl. that's what i have written the first term above. why i did is because the question says the group must contain minimum 1 teacher and max it can contain only two teachers since u have only two teachers out of the 12 ppl.   so my 2nd term has 2 teachers and one from the left lot. i hope u were able to understand a little bit.