# Statistics, trouble grasping logic

I'm not sure how to solve the questions in my math class because I can't grasp the logic behind the story.

For example given the following questions:

There are 8 men and 6 women. 4 Men and 2 women will be selected to form a team, in how many ways can the teams be arranged?

I would answer the above question with the following solution: $\binom{8}{4} * \binom{6}{2} = \frac{8!}{(4! \centerdot 4!)} * \frac{6!}{(4! \centerdot 2!)} = 1050$. Which I think is correct.

But with problems like:

In a competition with 10 competitors, in how many ways can 3 medals be shared among them?

I would answer this question the same way I answered the above question with: $\binom{10}{4}$. To my surprise this seems to be incorrect, it's probably because I can't get a hold on the logic involved.

It would be great if the logic behind the problems can be explained in some detail.

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If the three medals are identical (and if you replace 10-choose-4 by 10-choose-3), your answer is correct. But if one distinguishes between the medals (gold, silver and bronze), there are 6 times more different configurations. –  Did Apr 12 '12 at 9:06
Thus, there are $10*9*8 = \frac{10!}{7!}$ ways to distribute three medals between 10 competitors (e.g. golden medal goes to #5, silver medal goes to #7, bronze medal goes to #2), but only $\frac{10*9*8}{3!} = \binom{10}{3}$ ways to choose what three competitors will receive the medals (e.g. #2, #5 and #7 are getting some medals / positions / etc).