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A senate committee is to be formed from 51 Republican and 49 Democratic Senators.

The chair and vice chair must be Republicans and the two remaining members must
be Democrats, how many possible committees are there?

User Kunu
by
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1 Answer

2 votes
To determine the number of possible committees that can be formed with the given conditions, we can break down the process step by step:

Step 1: Selecting the chair from the Republican Senators
Since the chair must be a Republican, we have 51 options to choose from.

Step 2: Selecting the vice chair from the remaining Republican Senators
After selecting the chair, there will be 50 Republican Senators remaining. The vice chair must also be a Republican, so we have 50 options to choose from.

Step 3: Selecting the two remaining members from the Democratic Senators
We have 49 Democratic Senators, and we need to select two of them to be the remaining members. The order of selection does not matter, so we can use a combination formula. The number of ways to choose two Democrats from a group of 49 is given by:

C(49, 2) = 49! / (2!(49-2)!) = 49! / (2! * 47!) = (49 * 48) / 2 = 1,176

Step 4: Calculating the total number of possible committees
To calculate the total number of possible committees, we multiply the results from each step:

Total number of committees = Number of options for chair * Number of options for vice chair * Number of ways to choose two Democrats
Total number of committees = 51 * 50 * 1,176 = 3,690,000

Therefore, there are 3,690,000 possible committees that can be formed with a chair and vice chair being Republicans, and the two remaining members being Democrats.
User Goutham Ganesan
by
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