The chart above shows political and geographic data on a certain

Question

https://gmatclub.com/forum/download/file.php?id=73611 The chart above shows political and geographic data on a certain legislative committee of 20 members, each of whom belongs to 1 of 2 political parties and lives in 1 of 4 regions. How many subcommittees of this legislative committee are possible that contain exactly 1 member from each of the 4 regions and the same number of members from each of the 2 political parties? A. 10 B. 20 C. 99 D. 246 E. 495 2023-12-05_20-53-06.png 2023-12-05_20-52-23.png

Bunuel · Accepted Answer

Deconstructing the wording, we find that we need subcommittees with one member from each of the four regions, resulting in  exactly four members , of which  two are from Party A and two from Party B .

Since Party B has fewer members and they reside in fewer regions, let's construct the subcommittees starting with Party B first. As Party B must delegate  one member from each region  it is represented in, we have the following scenarios for Party B's delegation:

1. From North and South;
2. From North and East;
3. From South and East.

Let's count the possibilities:

1. Party B delegates one member from North and South: this can be done in 2*4 = 8 ways. In this case, Party A should delegate from West and East: this can be done in 3*2 = 6 ways. Total for this case = 8*6 = 48;

2. Party B delegates one member from North and East: this can be done in 2*3 = 6 ways. In this case, Party A should delegate from South and West: this can be one in 1*3 = 3 ways. Total for this case = 6*3 = 18;

3. Party B delegates one member from South and East: this can be done in 4*3 = 12 ways. In this case, Party A should delegate from North and West: this can be done in 5*3 = 15 ways. Total for this case = 12*15 = 180.

Total = 48 + 18 + 180 = 246.

Answer: D.

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=81993 ___ Harsha GMAT-Club-Forum-jrqvp3hn.png

gmatophobia · Answer

A few points to note:

We need the same number of members from each of the 2 political parties. Hence, out of the four members, we need two members of party A and two members of party B. 
 We don't have any members of Party B in the West region. Hence, the member selected from the western region will belong to Party A. 
 So, we need to select only one member of party A who can be either from the North, South, or East region. Once, the region of the member from Party A is identified, the remaining two regions will have the member from Party B. 
For example, if the member from the north region belongs to Party A, then, the members from the remaining two regions i.e. South and East must belong to Party B

Hence, we can have the following combinations

(West → Party A & North → Party A) (East → Party B & South → Party B) 
  (West → Party A & South→ Party A) (East → Party B & North → Party B) 
  (West → Party A & East → Party A) (North→ Party B & South → Party B)

Required combinations

(West → Party A & North → Party A) (East → Party B & South → Party B) =  ^3C_1 * ^5C_1 * ^3C_1 * ^4C_1 = 180   
  (West → Party A & South→ Party A) (East → Party B & North → Party B) =  ^3C_1 * ^1C_1 * ^3C_1 * ^2C_1 = 18   
  (West → Party A & East → Party A) (North→ Party B & South → Party B) =  ^3C_1 * ^2C_1 * ^4C_1 * ^2C_1 = 48

Total possible combinations = 180 + 18 + 48 = 246

Option D

DanTheGMATMan · Answer

Write out the possible combinations of regions for each party and then put the numbers in. Light work:

https://www.youtube.com/watch?v=2JoLZAIpOXY

KarishmaB · Answer

A quick tree diagram will help here. They can be quite useful. Note that West has only 3 Party A people (0 party B people) so 1 of those must be selected. This gives us 3 ways. Next, go to South - either 1 of Party A or 1 of the 4 party B can be selected. If party A is selected from South, then from north and east, party B candidates have to be selected and so on... Screenshot 2024-06-22 at 4.37.43 PM.png Total possible subcommittees = 18 + 48 + 180 = 246 Answer (D)

SergejK · Answer

As we have no overlaps, neither in the party affiliation nor the area that they live in, each individual lives in exactly one location and belongs to exactly one party. We see that if we add together party A and B to get 20 or all 4 regions are equal to 20. As we are asked, how many subcommittees of this legislative committee are possible that contain exactly 1 member from each of the 4 regions and the SAME (equal) number of members from each of the 2 political parties, we need to account for all possibilities. So what does this question mean? We need to have each region present. However, we have a constraint: the same number of members from each party, meaning we need 2 from each. So one of our possible combination could look something like this: 5C1*1C1*0C1*3C1. However, the constraints of the task don't actually allow us to do this as then we wouldn't have 4 people in total. As in party B not all regions are available, we best start with party B and provide 3C2 possible combinations for party B (3) instead of 4C2 (6) if all regions were filled. Now, if we choose 2 regions in party B, we need to choose the not chosen 2 for party A. So our combinations are (leaving out combination notation as xC1=x): 2*4*3*2+2*3*1*3+4*3*5*3=246

anushree01 · Answer

what a hard Qn is this! Firstly the language & secondly the method
Doing it under time pressure is a big thing!!!

ItsMeHudd · Answer

Thank you for the in depth response, my question is how do you recognize when to multiply and when to add your possible combinations?

egmat · Answer

Hi ItsMeHudd,

The solutions above may look different but all run on one rule.

Multiply = "AND" (choices that all happen together).  You multiply when a sequence of choices must  all  happen to build one complete outcome.

Take Case  3  above, Party B from South and East, Party A from North and West:

- 1 of  5  from North  AND  1 of  3  from West  AND  1 of  4  from South  AND  1 of  3  from East
- All four picks are needed for one subcommittee:  5 · 3 · 4 · 3 = 180

Add = "OR" (separate, mutually exclusive scenarios).  You add when you have different non-overlapping scenarios and any one of them counts as a valid outcome. You are not doing them together, you are listing alternatives.

That is why the three cases get added: Party A comes from North,  OR  from South,  OR  from East. They cannot overlap, since only one Party-A region is possible per subcommittee. So  180 + 18 + 48 = 246 .

The quick test.  Ask: to finish one outcome, do I still need to make another choice? Yes means  multiply . Is this instead a completely separate route to a valid outcome? Then  add .

A tiny parallel.  With  3  shirts and  2  pants, one outfit needs a shirt  AND  pants:  3 · 2 = 6  outfits. But to wear one item on your head, owning  3  hats  OR2  caps gives  3 + 2 = 5  choices. "Together" multiplies, "instead" adds.

Answer: D

AndyRugger · Answer

Bunuel's scenario-based solution is very elegant. I totally botched this question when I first saw it. Here's a possible "smart guessing" testing strategy to get closer to the actual answer if you can't break it down by scenario, and if you're unwilling/unable to skip the question.

Ignore the second stipulation for the subcommittee. What about the first: How many ways are there to create a subcommittee composed of exactly one member from each region? Well, you have four spaces on the subcommittee, each to be filled with a regional representative: 7 choices for the North, 5 for the South, 3 for the West, and 5 for the East. Total option is their combined multiple, which is 525.

So you know the answer must be less than 525 (which, unfortunately, is all of them). However, if you put on your Data Insights hat, you might realize that slightly less than half the total of 525 options will have equal representation (two members of each party) because the probability of selecting a person from either Party A or B is close to 50%. That puts you in the neighborhood of 250-ish for the answer, so 246 is the closest. Answer D.

The chart above shows political and geographic data on a certain

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