To determine the number of ways to choose 2 seniors from a group of 7 to go to an all-district meeting, we need to decide if this is a permutation or a combination.
### Step 1: Understand the difference between permutations and combinations
- Permutations: Used when the order of selection matters. For example, if selecting different positions such as President and Vice-President, the order in which the individuals are selected is important.
- Combinations: Used when the order does not matter. If just selecting a group, the order of selection is irrelevant.
Here, we are simply selecting 2 students to attend a meeting, and the order in which they are chosen doesn't matter. Therefore, this is a problem involving combinations.
### Step 2: Select the appropriate formula
For combinations, the formula for choosing [tex]\( r \)[/tex] items from [tex]\( n \)[/tex] items is:
[tex]\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items.
- [tex]\( r \)[/tex] is the number of items to choose.
- [tex]\( ! \)[/tex] denotes factorial, which means the product of all positive integers up to that number.
### Step 3: Substitute the given values into the formula
- Total number of seniors, [tex]\( n = 7 \)[/tex]
- Number of seniors to be chosen, [tex]\( r = 2 \)[/tex]
So, we substitute [tex]\( n = 7 \)[/tex] and [tex]\( r = 2 \)[/tex]:
[tex]\[
\binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2! \cdot 5!}
\][/tex]
### Step 4: Simplify the factorials
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
However, since [tex]\( 5! \)[/tex] is in both the numerator and the denominator, they cancel out:
[tex]\[
\frac{7 \times 6 \times 5!}{2! \times 5!} = \frac{7 \times 6}{2!}
\][/tex]
And [tex]\( 2! \)[/tex] is:
[tex]\[
2! = 2 \times 1 = 2
\][/tex]
So the calculation becomes:
[tex]\[
\frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21
\][/tex]
### Final Answer
Therefore, the number of ways to choose 2 seniors from 7 is given by the combination, which is [tex]\(\boxed{21}\)[/tex].
The correct answer is: C. Combination; number of ways [tex]\( = 21\)[/tex].
