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
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.
