Know 12 of 20. Want 8, 9, or 10 on list of 10.

1.

Ok, well the chances of knowing all 10 are

(12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13)*(4/12)*(3/11)

The chances

of knowing 9 of the 10 are

(12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13)*(4/12)*(the chance of not knowing the 10th question), where the chance of not knowing the 10th question is (8/11). Then you have to take into account that this can happen in 10 different ways - because there are 10 different choices for the question that the student gets wrong - (ie. it could be the first OR the second OR the third.. etc...) So you multiply your answer by 10.

The chances of knowing 8 of 10 are similarly constructed:

(12/20)*(11/19)*(10/18)*(9/17)*(8/16)*(7/15)*(6/14)*(5/13) * (chances of not knowing the last 2), where the chances of not knowing the last two are (8/12)*(7/11).

Now the two questions answered wrongly can be any of the 10, and you use your 10C2 button on your calculator to get this. (incidentally - you could have done this in the previous part - 10C1 = 10, which is what we got).

So now you have the chances of getting exactly 8, exactly 9 or exactly 10.

So in order to find the chances of getting AT LEAST 8, you just add up the previous three answers.