Question

The varsity croquet team, made up of 4 boys and 8 girls, travels to a game. Their coach, Tiss Teak, will take 7 of them in her station wagon. If they get into cars at random, find the following:
a.)how many possible groups of 7 can Miss Teak have in their car?
b.)How many groups would have 2 boys and 5 girls?
c.)What is the probability that her car has a group of 2 boys and 5 girls?
d.)What is the probability that her car has a group of all girls?
e.)What is the probability that her car has a group of all boys?
f.)What is the probability that Mark Wright and Al Pine, two of the boys, are in her car?

Answer 1

Well, for a, it's just \(\binom{12}{7}\).

b. \[\binom{4}{2}\cdot\binom{8}{5}\]c.\[{\binom{4}{2}\cdot \binom{8}{5}} \over \binom{12}{7}\]

Answer 2

is that a combination?

Answer 3

d.\[\binom{8}{7} \over \binom{12}{7}\]e.Less than 7 boys, so the probability is 0.
f.\[\binom{10}{5} \over \binom{12}{7}\]

Answer 4

And yes, those are all combinations.

Answer 5

If there's any one you think you need an explanation for, please say so.

Answer 6

Y do you divide certain ones?

Answer 7

And how come you multiply?

Answer 8

It's asking for the probability of a certain scenario, so you find the number of possible ways for that scenario, and divide it by the total number of scenarios.

Answer 9

And the multiplying?

Answer 10

So for b, it asks you to choose 2 boys, and 5 girls, so you need to find \(\binom{4}{2}\) for the number of ways to pick the boys, and \(\binom{8}{5}\) for the number of ways to choose the girls. Since these are exclusive, you have to multiply them together to find total number of ways to pick 2 boys and 5 boys.

Answer 11

Oh, so when they ask you for how many groups youou can make from the two categories, then you multiplyy the groups you get from just one category by the other?

Answer 12

Basically. You have to be careful that the groups are exclusive though. If some people belong to more than one group, it isn't that simple.

Answer 13

How does the situation or the equation change when some people belong to more than one group?

Answer 14

The simplest way is to do a case by case analysis then. For example, say you have 2 people that belong to both groups. Then you find how many ways it can happen if none of those two people are chosen, how many ways only one is chosen, and how many ways both of them are chosen. Once you have those just add them up.

Answer 15

Woah, so then how do you tell if you have to do permutation instead of combination?

Answer 16

It's permutations if order matters. A standard example would be a combination to a safe. If you have a safe where you know the combination is 3 digits long, and digits can't be repeated, it you would use permutations. If you tried to use combinations instead, then 123, 132, 213, 231, 312, 321 would all look the same. Permutations differentiate between these different possibilities.

In short, if order matters, use permutations, if order doesn't matter, use combinations.

Answer 17

So how do you differentiate between needing the order and how do you know which numbers to use, like in the question?

Answer 18

Oh, and how did you get letter d???

Answer 19

An to thnk about it, I really don't get letter f, either...

Answer 20

If you need the order, you can explicitly list the objects as "the first one" "the second one" "the third one" and so forth.

Answer 21

As for d, we have to find the number of ways we can have a group of all girls. That's given by \(\binom{8}{7}\) since we have 8 girls, and we have to choose 7 of them. But since it's asking for the probability, we have to divide by the total number of ways we can pick 7 people which is \(\binom{12}{7}\). So the probability of a group of 7 girls is\[\binom{8}{7} \over \binom{12}{7}\]

Answer 22

And then for f, we have to assume that Mark and Al are already chosen. So we need to choose 5 more people to get 7 people total. Also, you started with 12 people but since you've already chosen 2 of them, you have \(\binom{10}{5}\) ways to choose Mark and Al. Then you have to divide by the total again to find the probability.

Answer 23

Why assume Mark and Al are already chosen?Wouldn't it be the other way around?

Answer 24

It asks for the probability that they were chosen to be in her car, so to find the number of ways they could have been chosen to be in her car you have to assume they were chosen.

Answer 25

So then y arre they coounted out? Im so confused about this one...

Answer 26

They aren't counted out, we just say that they've already been chosen. Since there are seven seats in the car, and two of them are already taken by Mark and Al, we only have 5 seats left to choose from.

Answer 27

That is confusing. Im squeezing my brain juices out to try and think that abstract!!!^O^

Answer 28

THANK YOU SO MUCH KingGeorge. I believe you are one of the finest people I know mathwise. Look out for this last question I am going to ask! <3

Answer 29

Thanks for the compliment :)