A local card club will send 3 representatives to the national conference. If the the local club has 8 members, how many different groups of representatives could the club send?

Answer

To find the number of different groups of representatives that the local card club can send to the national conference, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Identify the total number of members and the number of representatives to be chosen.
We have a total of 8 members in the club, and we need to choose 3 representatives.

### Step 2: Use the combination formula.
The number of ways to choose \( r \) representatives from \( n \) members is given by the combination formula:
\[ C(n, r) = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 8 \) and \( r = 3 \).

### Step 3: Substitute the values into the formula.
\[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \]

### Step 4: Simplify the factorials.
We can expand \( 8! \) as follows:
\[ 8! = 8 \times 7 \times 6 \times 5! \]
Now substituting this back into the combination formula:
\[ C(8, 3) = \frac{8 \times 7 \times 6 \times 5!}{3! \cdot 5!} \]
The \( 5! \) in the numerator and denominator cancels out:
\[ C(8, 3) = \frac{8 \times 7 \times 6}{3!} \]

### Step 5: Calculate \( 3! \).
\[ 3! = 3 \times 2 \times 1 = 6 \]

### Step 6: Substitute \( 3! \) back into the equation.
\[ C(8, 3) = \frac{8 \times 7 \times 6}{6} \]

### Step 7: Simplify the expression.
Now we can simplify:
\[ C(8, 3) = 8 \times 7 = 56 \]

### Conclusion:
Thus, the number of different groups of representatives that the club could send is \( \boxed{56} \).

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