https://file.aiquickdraw.com/mathbot/file/17273032593959ripb4hc.jpg

The problem asks how many different committees of 40 senators can be formed if the two senators from the same state are considered identical, assuming that there are 50 states in total.

### Solution:

#### Step 1: Total Senators and Identical Senators per State
- There are 2 senators per state, and a total of 50 states, which gives us 100 senators in total.
- Since senators from the same state are considered identical, we treat each state's two senators as a single entity. This simplifies the problem to choosing committees from 50 "senator pairs" (one per state).

#### Step 2: Number of Possible Committees
- The problem reduces to selecting 40 pairs from the 50 available states. This is a combination problem because the order of selection does not matter.
- The number of ways to choose 40 pairs from 50 is given by the combination formula:
  \[
  \binom{50}{40} = \frac{50!}{40!(50-40)!} = \frac{50!}{40!10!}
  \]

#### Step 3: Compute \(\binom{50}{40}\)
Since \(\binom{50}{40} = \binom{50}{10}\) (by the symmetry property of combinations), the final result is:
\[
\binom{50}{10} = \frac{50!}{10!40!}
\]

We can compute this using a calculator for the exact value.

Would you like me to calculate the exact number or provide more details?

### Related Questions:
1. How would the solution change if each senator was distinct (i.e., not considering state pairs)?
2. What if the committee was of a different size, say 30 senators, instead of 40?
3. How would the solution differ if there were 60 states but the same pairing rule applied?
4. Can you explain the significance of the combination formula \(\binom{n}{k}\) in choosing subsets?
5. How would the outcome change if there was a restriction that at least one senator from each state must be chosen?

### Tip:
Always check if the problem involves distinct entities (like people) or if some entities are considered identical—this can significantly change the approach to combinatorics problems.

How many different committees of 40 senators can be formed if the two senators from the same state (50 states in all) are considered identical?

Explore how to calculate the number of different committees of 40 senators that can be formed when senators from the same state are considered identical. This problem involves combinatorics and uses the combination formula and symmetry property. Learn how to solve this step-by-step.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation