Basic, but confusing, counting problem

[Bre Ntt]

This is a pretty basic counting problem, but it is confusing me to no end. I know the answer (from the back of the book), but I just don't understand the answer.

Homework Statement

Find the probability of getting exactly 4 numbers correct in a lottery where 6 numbers are chosen from 49 numbers (no repetitions.)


The answer is (43 Choose 2)(6 Choose 4)/(49 Choose 6)

(43 Choose 2)(6 Choose 4) is apparently the number of possible tickets with exactly 4 correct answers. But I'm not sure exactly why. I think (43 Choose 2) is the number of possible tickets which have 2 numbers which do not match the winning numbers. Why multiply this by (6 Choose 4) though? I'm thinking it has something to do with considering how many ways you can choose 4 numbers from the 6 winning numbers, but I don't understand why one multiplies it by the number of ways possible tickets which have exactly 2-non-matching numbers.


I can't figure out an intuitive way to grasp this counting process. It almost makes sense to me, but not quite. Can anyone explain this with an intuitive counting argument?

 

[Simon Bridge]

Imagine there are 49 balls in a bag, 6 are red and 43 are blue.
You pick six balls out of the bag.
What are the odds you'll get exactly 4 red balls.
Now do you get it?

There are 43 numbers that are incorrect and six correct numbers, total 49.
2 of your numbers come from the 43 incorrect ones.
4 of your numbers come from the 6 correct ones.

Have a look at:
http://betterexplained.com/articles/easy-permutations-and-combinations/

 

[Bre Ntt]

That way of thinking about it helps. Thank you :)