Question 1210159: In the United States, there are 50 states. Each state is represented by 2 senators. In how many ways can we form a committee with 5 senators, in which no two of the senators are from the same state?
Found 3 solutions by CPhill, greenestamps, math_tutor2020:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break this problem down step by step.
**1. Total Number of Senators**
* There are 50 states, and each state has 2 senators, so there are $50 \times 2 = 100$ senators.
**2. Choosing the States**
* We need to form a committee of 5 senators, and no two senators can be from the same state.
* First, we need to choose 5 states out of the 50 states. The number of ways to do this is $\binom{50}{5}$.
**3. Choosing the Senators from Each State**
* For each of the 5 chosen states, we have 2 senators to choose from.
* Since we need to choose one senator from each of the 5 states, we have 2 choices for each state.
* Therefore, we have $2^5$ ways to choose the senators from the selected states.
**4. Calculating the Total Number of Ways**
* To get the total number of ways to form the committee, we multiply the number of ways to choose the states by the number of ways to choose the senators from those states.
Total ways = $\binom{50}{5} \times 2^5$
**5. Performing the Calculations**
* $\binom{50}{5} = \frac{50!}{5!45!} = \frac{50 \times 49 \times 48 \times 47 \times 46}{5 \times 4 \times 3 \times 2 \times 1} = 2,118,760$
* $2^5 = 32$
Total ways = $2,118,760 \times 32 = 67,800,320$
**Therefore, there are 67,800,320 ways to form a committee with 5 senators, in which no two of the senators are from the same state.**
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Choose 5 of the 50 states in C(50,5) = 2118760 ways.
From each of those 5 states, choose 1 of the 2 senators in (C(2,1)^5) = 2^5 = 32 ways.
Total number of ways of choosing the committee: 2118760*32 = 67800320.
ANSWER: 67800320
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
A similar question is found here
I recommend checking it out and using it as a tool to solve your current problem. If you're still stuck, then read on.
There are 50*2 = 100 senators total.
For the first slot we have 100 choices.
For the second slot we have 100-2 = 98 choices.
We do not have 99 choices since we exclude the second senator from whatever state was chosen as the first slot.
Otherwise, this committee could potentially have two senators from the same state.
We keep this process of subtracting 2 until we account for all five slots.
The five slots have these choices: 100, 98, 96, 94, 92
If order mattered then we would have 100*98*96*94*92 = 8,136,038,400 different committees.
Roughly 8.136 billion.
However, it appears that none of the seats have a title (such as "chairman" or "treasurer"), so this means a committee like ABCDE is the same as ACBDE.
Order does not matter.
How many ways are there to scramble the letters in ABCDE?
There are 5 letters and 5*4*3*2*1 = 120 permutations.
So we must divide that 8.136 billion figure by 120 to arrive at the correct final answer.
8136038400/120 = 67800320
When using commas to make the number more readable, it would look like this 67,800,320. This is approximately 67.8 million.
A bit of random trivia: If you were to form a different committee every second, then it would take about 784.7 days (a little over 2 years) for you to try all 67.8 million options.
Answer: 67800320
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