Please check my work on this factorial question?

Question

So if there's a group of 7 girls and 9 guys, how many ways can a group of 5 people be selected if at least one is female?

if there's one girl, I got 882 possibilities
for 2 girls, I got 1764 possibilities
for 3 girls, I got 1260 possibilities
for 4 girls, I got 315 possible
for 5 girls, I got 21 possible

adding those up, I got 4,242 possible. Please double check. If I'm wrong, then help explain.

As a second part, how many possible if you have to have at least 1 girl and one guy in the group? For this one I just subtracted the 21 from 4,242 and got 4,221.

Thanks in advance!

ok pickles, you jackass...

If you knew anything about factorials you'd see that I already did all the work, I just wanted someone to check my logic. Moron.

thanks brambly, you're awesome...

Answer 1

There's a simpler way to do this.
First, figure out how many ways you can choose 5 people out of 16. That's 16!/(11!5!), or 4368
Then figure out how many ways you can choose the five people so that all five are male. That's 9!/(5!4!), or 126.
Subtract 126 from 4368 and you get 4242.

Since I got the same answer as you, using a different method, that should be correct.

The 4221 is also correct, since 21 is indeed the number of ways you can pick an all-female group.

Answer 2

::

Part 1. Your approach is correct. Another way of doing it: at least one is female is the same as all combinations minus none is female, or all is male.

All combinations: 16!/(5!11!) = 4,368

All male: 9!/(5!4!) = 126

At least one female: 4,368 - 126 = 4,242


Part 2. Also correct. Alternative is

At least one male, one female = All combination - (all female + all male)

All female = 7!/(5!2!) = 21

At least one male, one female = 4,368 - (21 + 126) = 4,221

Answer 3

Check it youself like I did when I was in school and quit getting people on the Internet to do your work for you.