A committee that includes 6 members is about to be divided into 2 subcommittees with 3 members each. On what percent of the possible subcommittees that Michael is a member of is David also a member?
A, 10%
B, 20%
C, 25%
D, 40%
E, 50%
Source: Master GMAT Practice Test (free)
Solution:
Visualize the problem thus: put 3 people in room A and the other 3 in room B. Say Michael is in room A, thus room A has 2 more open places out of 5 (2 in Michael's room plus 3 in the other room) - the chance that David will be in one of the 2 places in Michael's room are 2/5 = 40%
I have gone through the solution provided above but I want a more standard way of solving this problem using Combination formula. Tricks as those suggested here may not be applied to every problem.
Selecting a Committee
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3-member committees that include Michael:kartikshah wrote:A committee that includes 6 members is about to be divided into 2 subcommittees with 3 members each. On what percent of the possible subcommittees that Michael is a member of is David also a member?
A, 10%
B, 20%
C, 25%
D, 40%
E, 50%
Source: Master GMAT Practice Test (free)
Solution:
Visualize the problem thus: put 3 people in room A and the other 3 in room B. Say Michael is in room A, thus room A has 2 more open places out of 5 (2 in Michael's room plus 3 in the other room) - the chance that David will be in one of the 2 places in Michael's room are 2/5 = 40%
I have gone through the solution provided above but I want a more standard way of solving this problem using Combination formula. Tricks as those suggested here may not be applied to every problem.
There are two spaces to fill on this committee and 5 people left.
Number of options = 5C2 = 10.
3-member committees that include Michael and David:
There is one space to fill on this committee and 4 people left.
Number of options = 4.
(committees with both Michael and David)/(committees with Michael only) = 4/10 = 40%.
The correct answer is D.
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Hi Mitch,
Thanks for posting the solution. I was wondering what if the question was modified slightly to read as below:
There are 6 members on a committee that is to be subdivided into two committees of 3 members each. Michael and David are two of the six members. What is the possibility that they are always together in the same sub-committee?
Would the possibility be worked out as under:
Total ways of forming the sub-committees = 6C3 = 20 ways
Total ways of forming a sub-committee with M+D+one more member = 4C1 = 4 ways
Possibility = 4/20 = 20%
Thanks for clarifying.
Thanks for posting the solution. I was wondering what if the question was modified slightly to read as below:
There are 6 members on a committee that is to be subdivided into two committees of 3 members each. Michael and David are two of the six members. What is the possibility that they are always together in the same sub-committee?
Would the possibility be worked out as under:
Total ways of forming the sub-committees = 6C3 = 20 ways
Total ways of forming a sub-committee with M+D+one more member = 4C1 = 4 ways
Possibility = 4/20 = 20%
Thanks for clarifying.
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Yes. You have done the modified question correctly.kartikshah wrote:Hi Mitch,
Thanks for posting the solution. I was wondering what if the question was modified slightly to read as below:
There are 6 members on a committee that is to be subdivided into two committees of 3 members each. Michael and David are two of the six members. What is the possibility that they are always together in the same sub-committee?
Would the possibility be worked out as under:
Total ways of forming the sub-committees = 6C3 = 20 ways
Total ways of forming a sub-committee with M+D+one more member = 4C1 = 4 ways
Possibility = 4/20 = 20%
Thanks for clarifying.
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If the 6 people are being ASSIGNED to subcommittees A and B, each with 3 members:kartikshah wrote:Hi Mitch,
Thanks for posting the solution. I was wondering what if the question was modified slightly to read as below:
There are 6 members on a committee that is to be subdivided into two committees of 3 members each. Michael and David are two of the six members. What is the possibility that they are always together in the same sub-committee?
Would the possibility be worked out as under:
Total ways of forming the sub-committees = 6C3 = 20 ways
Total ways of forming a sub-committee with M+D+one more member = 4C1 = 4 ways
Possibility = 4/20 = 20%
Thanks for clarifying.
The probability that Michael and David are both assigned to subcommittee A is 20%.
The probability that Michael and David are both assigned to subcommittee B is 20%.
The probability that Michael and David are assigned to the SAME subcommittee -- either A OR B -- is 40%.
The key here is that the 6 people are being DIVIDED into TWO groups of 3.
Since there are TWO groups of 3, there are TWICE as many ways to get a favorable outcome.
Here are all of the ways to DIVIDE 6 people M, D, W, X, Y and Z into TWO groups of 3:
MDW - XYZ
MDX - WYZ
MDY - WXZ
MDZ - WXY
MWX - DYZ
MWY - DXZ
MWZ - DXY
MXY - DWZ
MXZ - DWY
MYZ - DWX
In the 4 red groupings, Michael and David are together.
Since there are 10 total ways to DIVIDE the 6 people:
P(Michael and David are in the same group) = 4/10 = 40%.
Using combinatorics:
The number of ways to DIVIDE 6 people into two groups of 3 = 6C3/2! = 10.
The number of ways to form a group of 3 that includes both Michael and David = 4.
P(Michael and David are in the same group) = 4/10 = 40%.
There is a distinction between ASSIGNING people to groups and DIVIDING them into groups.
For more on this distinction, check my posts here:
https://www.beatthegmat.com/combination-t92104.html
https://www.beatthegmat.com/forming-teams-t73034.html
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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