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14 Oct 2023, 11:39
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D
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67% (03:05) correct
33%
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wrong
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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
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05 Dec 2023, 10:47
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yrozenblum
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.
Updated on: 14 Feb 2025, 21:59
Originally posted by HarshavardhanR on 16 Apr 2024, 00:16.
Last edited by HarshavardhanR on 14 Feb 2025, 21:59, edited 1 time in total.
Last edited by HarshavardhanR on 14 Feb 2025, 21:59, edited 1 time in total.
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