A delegation from Gotham City goes to Metropolis to discuss a limited Batman-Superman partnership. If the mayor of Metropolis chooses 3 members of the 7-person delegation to meet with Superman, how many different 3-person combinations can he choose?

Answer

To solve the problem of how many different 3-person combinations can be chosen from a 7-person delegation, we will use the concept of combinations. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We need to choose 3 members from a total of 7 members. This is a combination problem because the order in which we choose the members does not matter.

### Step 2: Use the Combination Formula
The formula for combinations is given by:

\[ \binom{n}{r} = \frac{n!}{(n - r)! \cdot r!} \]

where:
- \( n \) is the total number of items (in this case, 7),
- \( r \) is the number of items to choose (in this case, 3),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

### Step 3: Substitute the Values into the Formula
Here, we need to calculate \( \binom{7}{3} \):

\[ \binom{7}{3} = \frac{7!}{(7 - 3)! \cdot 3!} = \frac{7!}{4! \cdot 3!} \]

### Step 4: Calculate the Factorials
Now we will calculate the factorials:

- \( 7! = 7 \times 6 \times 5 \times 4! \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

### Step 5: Substitute the Factorials Back into the Equation
Now we can simplify:

\[ \binom{7}{3} = \frac{7 \times 6 \times 5 \times 4!}{4! \cdot 3!} \]

The \( 4! \) cancels out:

\[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3!} = \frac{7 \times 6 \times 5}{6} \]

### Step 6: Simplify the Expression
Now we can simplify further:

\[ \binom{7}{3} = 7 \times 5 = 35 \]

### Conclusion
Thus, the number of different 3-person combinations that can be chosen from the 7-person delegation is **35**.

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