Question

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?

Answer 1

Let's say we have 3 slots A,B,C

how many do we have to choose from for slot A?

Answer 2

7?

Answer 3

Correct, then for slot B?

Answer 4

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

Answer 5

6?

Answer 6

yep, and then C?

Answer 7

5

Answer 8

multiply?

Answer 9

yes, you'll multiply 7, 6 and 5

Answer 10

210

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

the reason why is because there are 6 ways to order any three objects

xyz
xzy

yxz
yzx

zxy
zyx

Answer 15

210 divided by 3?

Answer 16

210/6 actually

Answer 17

or 210/(3!)

Answer 18

ah okay. I read your response wrong haha so 35

Answer 19

correct

Answer 20

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

Answer 21

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

Answer 22

that is correct

Answer 23

thank you

Answer 24

if you think of it in terms of 7 slots A through G we have this

Answer 25

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

Answer 26

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)

Answer 27

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

Answer 28

you're welcome