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A committee of three people is to be chosen from four teams

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A committee of three people is to be chosen from four teams [#permalink] New post  12 Apr 2012, 17:48
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A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A. 20
B. 22
C. 26
D. 30
E. 32



Show Spoiler
The easy way to solve this problem is to say there are 4C3 ways to choose a team. (2C1)(2C1)(2C1)(2C0) to choose the players. Hence the solution is (4C3)(2C1)(2C1)(2C1)(2C0)=32.

You can also do this longer way: Since each member on the committee must be from a different team lets look at the converse: How many ways can people of the same team be on the committe and subtract them out. We can pick 3 people from 8, 8C3=56, this is the total number of combinations. We can find the number of ways the a two people from the same team are on the committee by

1) The paired team on the committee is the first team
(2C2)(2C1)(2C0)(2C0)=2
(2C2)(2C0)(2C1)(2C0)=2
(2C2)(2C0)(2C0)(2C1)=2

2) The paired team on the committee is the second team

(2C1)(2C2)(2C0)(2C0)=2
(2C0)(2C2)(2C1)(2C0)=2
(2C0)(2C2)(2C0)(2C1)=2

3) The paired team on the committie is the third team

(2C1)(2C2)(2C0)(2C0)=2
(2C0)(2C2)(2C1)(2C0)=2
(2C0)(2C2)(2C0)(2C1)=2


4) The paired team on the committie is the fourth team
(2C1)(2C0)(2C0)(2C2)=2
(2C0)(2C1)(2C0)(2C2)=2
(2C0)(2C0)(2C1)(2C2)=2

All those two add up to 24. 56-24=32.

Here is my question, please help me answer below
TWO APPROACHES
COMBINATION==========================
Now if I wanted to do find the number of combinations of pair of the same team on the committee using combination.

(4C1)=# of ways to pick one team where we will grab two people from
(1C1)= # of ways to pick the two people from the same two people on the team
(3C1)= # of ways to pick the last person from the other three teams
(2C1)=# of ways to pick the person from the other group

Hence (4C1)(1C1)(3C1)(2C1)=24 . 56-24=32. Is my thinking correct here?

PERMUTATIONS================================

Now if I wanted to do find the number of combinations of pair of the same team on the committee using permutation.
I have three slots for the three committee members. I have 8 people. I have 8 to choose from for the first slot. 1 person for the second slot, as it has it be the first slots team member. The last person can be any of the 6. Hence 8*1*6=48. But I did not consider the different arrangement's of the Pair Pair Nopair committee. I have 3!/2!=3 to jumble them. So I have 48*3=144 ways of picking two people from the same team on the committe. 56-144 is negative! What am I doing wrong here !! HELP Bunuel!! Help Anyone!! Any help would be appreciated!!
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  12 Apr 2012, 21:50
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alphabeta1234 wrote:
A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A)20
B)22
C)26
D)30
E)32


I'm not sure what are you exactly doing in your last approach.

Anyway, if you want to solve with slot method the simpler solution would be: 8*6*4/3!=32, 8 ways to choose for the first slot, 6 ways to choose for the second slot, 4 ways to choose for the third slot and dividing by 3! to get rid of duplication.

Another way to solve this problem would be: \(C^3_4*2^3=32\), where \(C^3_4\) is # of ways to choose which 3 team members out of 4 will be represented in the committee and multiplying this by 2*2*2 since we can choose any of 2 members from each chosen team.

Answer: E.

Check some VERY similar question to practice:
if-a-committee-of-3-people-is-to-be-selected-from-among-98533.html
ps-combinations-94068.html
ps-combinations-101784.html
committee-of-88772.html
if-4-people-are-selected-from-a-group-of-6-married-couples-99055.html
if-there-are-four-distinct-pairs-of-brothers-and-sisters-99992.html
a-committee-of-3-people-is-to-be-chosen-from-four-married-94068.html

Hope it helps.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  04 Oct 2012, 23:41
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A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A. 20
B. 22
C. 26
D. 30
E. 32

My Method

First we find out the amount of combinations of 3 person teams from a pool of 8. C(3/8) = 8!/5!*3! = 56 ways

Next I find the number of ways we CAN make a 3 person team using two from the same group, so

We take 2 people from a 2 person group C(2/2) and mutiply that by taking any 1 person from the remaining 6 C(1/6)

C(2/2)*C(1/6) = 6

This is the number of combinations by taking both parties from pair A. As we have a four pairs we must multiply this by 4 (4x6 = 24)

So there are 24 ways in which two people from the same pair can work together.

Finally we subtract this from the total number of combinations to find the number of groups where pairs do NOT work together....

56 - 24 = 32!
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  02 Oct 2012, 03:36
Bunuel wrote:
alphabeta1234 wrote:
A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A)20
B)22
C)26
D)30
E)32


I'm not sure what are you exactly dong in your last approach.

Anyway, if you want to solve with slot method the simpler solution would be: 8*6*4/3!=32, 8 ways to choose for the first slot, 6 ways to choose for the second slot, 4 ways to choose for the third slot and dividing by 3! to get rid of duplication.

Another way to solve this problem would be: \(C^3_4*2^3=32\), where \(C^3_4\) is # of ways to choose which 3 team members out of 4 will be represented in the committee and multiplying this by 2*2*2 since we can choose any of 2 members from each chosen team.

Answer: E.

Check some VERY similar question to practice:
if-a-committee-of-3-people-is-to-be-selected-from-among-98533.html
ps-combinations-94068.html
ps-combinations-101784.html
committee-of-88772.html
if-4-people-are-selected-from-a-group-of-6-married-couples-99055.html
if-there-are-four-distinct-pairs-of-brothers-and-sisters-99992.html
a-committee-of-3-people-is-to-be-chosen-from-four-married-94068.html

Hope it helps.



Hi Brunel

I did not inderstand the concept behind writing 4c3 it means selecting 3 person out of 4 but it is 4 team so it should be 8c3

Am i Correct

Regards]

Archit
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  02 Oct 2012, 07:49
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Archit143 wrote:
Bunuel wrote:
alphabeta1234 wrote:
A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A)20
B)22
C)26
D)30
E)32


I'm not sure what are you exactly dong in your last approach.

Anyway, if you want to solve with slot method the simpler solution would be: 8*6*4/3!=32, 8 ways to choose for the first slot, 6 ways to choose for the second slot, 4 ways to choose for the third slot and dividing by 3! to get rid of duplication.

Another way to solve this problem would be: \(C^3_4*2^3=32\), where \(C^3_4\) is # of ways to choose which 3 team members out of 4 will be represented in the committee and multiplying this by 2*2*2 since we can choose any of 2 members from each chosen team.

Answer: E.

Check some VERY similar question to practice:
if-a-committee-of-3-people-is-to-be-selected-from-among-98533.html
ps-combinations-94068.html
ps-combinations-101784.html
committee-of-88772.html
if-4-people-are-selected-from-a-group-of-6-married-couples-99055.html
if-there-are-four-distinct-pairs-of-brothers-and-sisters-99992.html
a-committee-of-3-people-is-to-be-chosen-from-four-married-94068.html

Hope it helps.



Hi Brunel

I did not inderstand the concept behind writing 4c3 it means selecting 3 person out of 4 but it is 4 team so it should be 8c3

Am i Correct

Regards]

Archit


4C3 is selecting 3 teams out of four. Then, since each team can give either of its two members for the committee, we should multiply 4C3 by 2*2*2.

Hope it's clear.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  02 Oct 2012, 11:31
Bunuel - What do you mean by selecting "3 teams" out of 4? The question states that 3 people form a committee (or team) and need to be selected from 4 teams of 2 people each. Archit is right. Shouldn't it be 8C3?

However the slotting method you mention is correct. 8*6*4/3!
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  03 Oct 2012, 02:07
Expert Reply
vsprakash2003 wrote:
Bunuel - What do you mean by selecting "3 teams" out of 4? The question states that 3 people form a committee (or team) and need to be selected from 4 teams of 2 people each. Archit is right. Shouldn't it be 8C3?

However the slotting method you mention is correct. 8*6*4/3!


There are 4 teams. Now, if we select 3 teams from those 4 and each will send one member then thee committee will have 3 members and no 2 members from the same team.

Hope it's clear.

P.S. Please follow the links in my post above for similar questions to practice.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  03 Oct 2012, 02:39
i think it should be only 4c3 * 2c1
3 teams out of four and 1 person out of 2
am i correct or missing sthn
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  03 Oct 2012, 02:43
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Archit143 wrote:
i think it should be only 4c3 * 2c1
3 teams out of four and 1 person out of 2
am i correct or missing sthn


EACH team out of 3 chosen can send ANY of its 2 members, so its 2*2*2*4C3.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  09 Mar 2016, 08:15
in the slot method why we divide by 3! to get rid of duplication? I am not sure about this concept. it would be great if any one explain
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  09 Mar 2016, 08:43
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bapun11 wrote:
in the slot method why we divide by 3! to get rid of duplication? I am not sure about this concept. it would be great if any one explain


It is an arrangement question similar to finding arrangements of AABBB = 5!/ (2!*3!) where 2! and 3! are done to remove the duplication of A and B as all As and all Bs are the same. Had all As or all Bs be different (A1 A2) or (B1 B2 B3) , then the answer would have been = 5!

Hope this helps.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  10 Mar 2016, 03:47
Here is how I solved this -

A/A1, B/B1, C/C1, D/D1

Total ways of choosing without any ristriction: 8C3 = 8!/3!5! = 56

Now total number of cases that are not required -

1. One couple is picked - 2C2
2. Remaining one member is picked from remaining 6 people - 6C1
3. Since there are 4 couple so this can be multiplied by 4.

total: 4*2C2*6C1 = 24

Hence, total number desired : 56-24 = 32
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  31 Oct 2017, 08:07
How I Solved it:

Teams with no couple chosen = Total no. of possible teams - Teams with one couple chosen
= 8C3 - 4C1x6C1
= 56 - 24 = 32

where:
4C1 --> From 4 couples choose any one
6C1 --> From the 6 remaining candidates, choose any one
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  20 Nov 2018, 07:22
Hi guys,

I have prepared a video with detailed explanation of this question.

Click here to see the explanation.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  23 Sep 2019, 10:48
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total pairs;; 8*6*4/3*2 ; 32
IMO E


alphabeta1234 wrote:
A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A. 20
B. 22
C. 26
D. 30
E. 32



Show Spoiler
The easy way to solve this problem is to say there are 4C3 ways to choose a team. (2C1)(2C1)(2C1)(2C0) to choose the players. Hence the solution is (4C3)(2C1)(2C1)(2C1)(2C0)=32.

You can also do this longer way: Since each member on the committee must be from a different team lets look at the converse: How many ways can people of the same team be on the committe and subtract them out. We can pick 3 people from 8, 8C3=56, this is the total number of combinations. We can find the number of ways the a two people from the same team are on the committee by

1) The paired team on the committee is the first team
(2C2)(2C1)(2C0)(2C0)=2
(2C2)(2C0)(2C1)(2C0)=2
(2C2)(2C0)(2C0)(2C1)=2

2) The paired team on the committee is the second team

(2C1)(2C2)(2C0)(2C0)=2
(2C0)(2C2)(2C1)(2C0)=2
(2C0)(2C2)(2C0)(2C1)=2

3) The paired team on the committie is the third team

(2C1)(2C2)(2C0)(2C0)=2
(2C0)(2C2)(2C1)(2C0)=2
(2C0)(2C2)(2C0)(2C1)=2


4) The paired team on the committie is the fourth team
(2C1)(2C0)(2C0)(2C2)=2
(2C0)(2C1)(2C0)(2C2)=2
(2C0)(2C0)(2C1)(2C2)=2

All those two add up to 24. 56-24=32.

Here is my question, please help me answer below
TWO APPROACHES
COMBINATION==========================
Now if I wanted to do find the number of combinations of pair of the same team on the committee using combination.

(4C1)=# of ways to pick one team where we will grab two people from
(1C1)= # of ways to pick the two people from the same two people on the team
(3C1)= # of ways to pick the last person from the other three teams
(2C1)=# of ways to pick the person from the other group

Hence (4C1)(1C1)(3C1)(2C1)=24 . 56-24=32. Is my thinking correct here?

PERMUTATIONS================================

Now if I wanted to do find the number of combinations of pair of the same team on the committee using permutation.
I have three slots for the three committee members. I have 8 people. I have 8 to choose from for the first slot. 1 person for the second slot, as it has it be the first slots team member. The last person can be any of the 6. Hence 8*1*6=48. But I did not consider the different arrangement's of the Pair Pair Nopair committee. I have 3!/2!=3 to jumble them. So I have 48*3=144 ways of picking two people from the same team on the committe. 56-144 is negative! What am I doing wrong here !! HELP Bunuel!! Help Anyone!! Any help would be appreciated!!
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  05 Feb 2020, 04:39
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alphabeta1234 wrote:
A committee of three people is to be chosen from four teams of two. What is the number of different committees that can be chosen if no two people from the same team can be selected for the committee?

A. 20
B. 22
C. 26
D. 30
E. 32

[/spoiler]


Solution:

There are 8 options for the first member, 6 options for the second member (since the first member and his/her teammate cannot be selected) and 4 options for the third member.

If the order were important (for instance, if we were selecting a president, a vice president and a secretary); there would be 8 x 6 x 4 options. However, as the order is not important, there are (8 x 6 x 4)/3! = 8 x 4 = 32 possible selections.

Alternate Solution:

First, let’s select three teams out of four. Since there are four teams and we are selecting three, this can be done in 4 ways (just choose one team which is not to be selected).

Next, out of the three teams we pick, we have two options for each team (the first member or the second member). Thus, there are in total 4 x 2 x 2 x 2 = 32 possible selections.

Answer: E
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  15 Feb 2020, 22:49
I have solved this question in this way, Lets say we have A, B, C and D teams with 2 members each. Our goal is to select 3 out of 4 which can be done in 4 ways ( ABC,BCD,CDA,DAB) . Now,lets take one way ABC, each team A or B or C have 2 members each

(1+1)--> a team (1+1)--> B team (1+1)--> C team

Again , all teams has to send so will multiple.

2 * 2 *2 --> 8 * 4(ways)--> 32 Answer.
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Re: A committee of three people is to be chosen from four teams [#permalink] New post  15 Feb 2020, 23:33
C43*C21*C21*C21=4*2*2*2=32

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