Question 1200688: In how many ways can the U.S senate select a chairperson, a treasurer and a secretary of a committee if all the senators are eligible for selection and no two senators from the same state can be selected together?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 940,800
Delete the comma if needed.
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Explanation:
To ensure we do not select two senators from the same state, we will first randomly select 3 states.
There are n = 50 choices and r = 3 selections.
Order does matter because each seat on the committee is named.
If each person on this committee had equal rank and the seats weren't named, then order wouldn't matter.
Since order matters, we turn to the nPr permutation formula.
P(n,r) = (n!)/( (n-r)! )
P(50,3) = (50!)/( (50-3)! )
P(50,3) = (50!)/( 47! )
P(50,3) = (50*49*48*47!)/( 47! )
P(50,3) = 50*49*48
P(50,3) = 117600
An alternative approach is this:- There are 50 choices for the first slot
- There are 49 choices for the second slot
- There are 48 choices for the third slot
Count down by 1 each time. That leads to 50*49*48 = 117600 different permutations.
There are 117,600 different ways to pick a trio of states from a pool of 50 where order matters.
After a state is chosen, there are 2 choices per state.
This is because each state sends 2 senators.
Therefore, any particular trio of states chosen will have 2*2*2 = 2^3 = 8 different senator permutations.
Example:
Let's say we picked {Delaware, Texas, Ohio} in that exact order.
Furthermore we'll have- senator A and senator B from Delaware
- senator C and senator D from Texas
- senator E and senator F from Ohio
For the subset {Delaware, Texas} we have the following 4 possibilitiesImagine a 2 by 2 table to generate these four permutations.
| senator C | senator D | | senator A | A,C | A,D | | senator B | B,C | B,D |
Now to include Ohio, we will have a table of 4 rows (representing those permutations listed above) and 2 columns.
That will give 4*2 = 8 different senator permutations of 3 senators chosen from those 3 states.
| senator E | senator F | | A,C | A,C,E | A,C,F | | A,D | A,D,E | A,D,F | | B,C | B,C,E | B,C,F | | B,D | B,D,E | B,D,F |
A code like B,D,E means - Senator B (from Delaware) is the chairperson.
- Senator C (from Texas) is the treasurer.
- Senator E (from Ohio) is the secretary.
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Summary:- There are 50*49*48 = 117600 ways to pick three states from a pool of 50 where order matters.
- There are 2^3 = 8 ways to pick three senators for any particular trio of states (one senator per state).
This must mean there are 117600*8 = 940,800 ways to form this committee.
Edit: The tutor @ikleyn offers the most efficient route. However, I'll stick to my route since this process could be useful in other areas.
Answer by ikleyn(52782)  (Show Source):
You can put this solution on YOUR website! .
Any of 100 senators can be elected to the 1st position.
Then any of remaining 98 eligible senators can be elected to the 2nd position.
Then any of remaining 96 eligible senators can be elected to the 3rd position.
In all, there are 100*98*96 = 940800 different options. ANSWER
Solved.
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