MHB In how many ways can a committee be formed....

[Raerin]

In how many ways can a 4-member committee be formed from a girl rep and a boy rep from each of grades 9, 10, 11, and 12?

a) There is no restriction?
I did 8C4 = 70, but apparently the correct answer is 28. I just want to confirm if my answer is right or not.

b) David and Ryan can not both be on the committee?
6C3 * 2C1 = 40 but, again, the apparent answer is 55.

c) Leigh and Sarah will either be on or off the committee?
I have no idea what to do for this restriction.

 

[soroban]

Hello, Raerin!

In how many ways can a 4-member committee
be formed from a girl rep and a boy rep from
each of grades 9, 10, 11, and 12 if:

(a) There is no restriction?
I did 8C4 = 70, but apparently the correct answer is 28.
I just want to confirm if my answer is right or not.

You are right . . . "They" are wrong.

This is evident in their answer to part (b).

(b) David and Ryan can not both be on the committee?
6C3 * 2C1 = 40 but, again, the apparent answer is 55.

Your answer is incorrect; they are right.

There are: ._8C_4 \,=\,70 possible committees.

If David and Ryan are both on the committee,
we must choose the other 2 from the other 6 people.
. . there are: _6C_2 = 15 ways.

Therefore, there are: .70 - 15 \,=\,55 ways
. . in which David and Ryan are not serving together.


*snicker*

(a) They claim there are 28 possible committees.
(b) They say 55 of them do not have both David and Ryan.

I guess it's true:
. . Five out of four people have trouble with marh.

(c) Leigh and Sarah will either be on or off the committee?
I have no idea what to do for this restriction.

If Leigh and Sarah are both on the committee,
choose 2 more from the other 6 people.
. . _6C_2 \,=\,15 ways.

If neither Leigh and Sarah are on the committee,
choose all 4 from the other 6 people.
. . _6C_4 \,=\,15 ways.

Therefore, there are: .15 + 15 \,=\,30 ways.

 

[Raerin]

Also, what if both (b and c) restrictions apply? Would it be 55+30? The answer says it's 23.

 

[I like Serena]

Also, what if both (b and c) restrictions apply? Would it be 55+30? The answer says it's 23.

With more restrictions the number can only go down...

Possibilities are:

1. David on, Ryan not, Leigh & Sarah on
2. David not on, Ryan on, Leigh & Sarah on
3. Neither David nor Ryan on, Leigh & Sarah on
4. David on, Ryan not, Leigh & Sarah not on
5. David not on, Ryan on, Leigh & Sarah not on
6. Neither David nor Ryan on, Leigh & Sarah not on

How many combinations in each case?