MEDAL!!
(suppose there is a group of 7 people from which we will make a committee)
In how many ways can we pick a three-person committee?

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- anonymous

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- jim_thompson5910

Let's say we have 3 slots A,B,C
how many do we have to choose from for slot A?

- anonymous

7?

- mathmate

Correct, then for slot B?

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## More answers

- mathmate

Recall that one of the sever is already in slot A.

- anonymous

6?

- jim_thompson5910

yep, and then C?

- anonymous

5

- anonymous

multiply?

- jim_thompson5910

yes, you'll multiply 7, 6 and 5

- anonymous

210

- jim_thompson5910

that won't be your final answer but it gets you closer

- jim_thompson5910

210 is the number of ways to pick 3 people IF order mattered
but there is no ranking on this committee, so order doesn't matter

- jim_thompson5910

so what you have to do is divide by 3! = 3*2*1 = 6 to get the correct count

- jim_thompson5910

the reason why is because there are 6 ways to order any three objects
xyz
xzy
yxz
yzx
zxy
zyx

- anonymous

210 divided by 3?

- jim_thompson5910

210/6 actually

- jim_thompson5910

or 210/(3!)

- anonymous

ah okay. I read your response wrong haha so 35

- jim_thompson5910

correct

- jim_thompson5910

another way is to use the combination formula
\[\Large _n C _r = \frac{n!}{r!*(n-r)!}\]
with n = 7 and r = 3 and you'll get the same answer

- anonymous

@jim_thompson5910 what if it was a four-person committee instead? wouldn't it be 7C4=35? also?

- jim_thompson5910

that is correct

- anonymous

thank you

- jim_thompson5910

if you think of it in terms of 7 slots A through G we have this
|dw:1441932178631:dw|

- jim_thompson5910

now imagine cutting a line through that group such that one side (say the left side) has 3 slots and the other side has 4 slots
|dw:1441932241059:dw|

- jim_thompson5910

arranging 7 people to go in slots A through C is the exact same as arranging the remaining 4 to go in D through G
so that's why 7 C 3 = 7 C 4
in general
\[\LARGE _n C _x = \ _n C _y\]
where x+y = n
(so in this case, x+y = 3+4 = 7 which is the value of n)

- anonymous

Yes, that is the way my instructor explains it also, thanks again!

- jim_thompson5910

you're welcome

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