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14 Oct 2023, 11:39
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68% (03:11) correct
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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
Attachment:
2023-12-05_20-53-06.png [ 21.77 KiB | Viewed 23312 times ]
Attachment:
2023-12-05_20-52-23.png [ 77.56 KiB | Viewed 16432 times ]
05 Dec 2023, 10:47
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yrozenblum
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
Attachment:
2023-12-05_20-53-06.png
Attachment:
2023-12-05_20-52-23.png
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.
14 Oct 2023, 13:45
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yrozenblum
Attachment:
Table.png
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
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
