Mathematics, 23.07.2020 18:01 emmina

Can someone help me, my homework is due Monday. If nobody answers correctly before Monday, my teacher will go over the problem in class and I will write the correct answer in the comments. Thank you to anyone who actually helps. There are 2 Senators from each of 50 states. We wish to make a 3-Senator committee in which no two members are from the same state. How many ways can we choose a Senator from a chosen state?

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Answer from: robert7248

$\huge\boxed{\text{2 ways}}$

Step-by-step explanation:

Intro:

We can use probability functions here to try and solve this problem.

We want to find how many ways to choose a senator from a chosen state. Note the "from a chosen state," meaning that it is not asking for how many ways to choose two senators from all 50 states but how to choose 1 from a chosen state (1 chosen state).

Let's calculate all possible values for each scenario.

Choosing a senator from one state:

We can either chose Senator A or Senator B for one state, since there are only two options available

This gives us a total of two possibilities to choose a senator from one state.

Choosing 3 states to be represented

The number of ways to choose 3 states being represented, we can use the combinations formula. We can write it as ₅₀C₃, which can be solved using the formula:

$\displaystyle \frac{n!}{(n-r)! \times r!}$

Where n is the total number of objects to choose from, and r is how many we want to choose.

Substituting in:

$\displaystyle \frac{50!}{(50-3)! \times 3!}$ $\displaystyle \frac{50!}{47! \times 3!}$ Since 47! will cancel out 50! from 47 downwards: $\displaystyle \frac{50\cdot 49\cdot 48}{3\cdot 2\cdot 1}$ $\displaystyle 19600$

Hence, there are 19,600 ways to choose 3 states from 50 total.

Choosing a 3 member committee

The total number of possible combinations to choose three senators from 50 states, none of the states being equal, we can multiply the number of ways to pick 3 states from 50 (19600) and multiply it by 2^3 since there are 3 members in the Senate and two Senators in each state.