To determine which expression represents the number of possible pairs of volunteers from an audience of 250 people, we need to use the concept of combinations. Combinations are used when the order of selection does not matter, and they are represented by the formula:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
where [tex]\( n \)[/tex] is the total number of people to choose from (in this case 250), and [tex]\( k \)[/tex] is the number of people to choose (in this case 2).
Let's break down the combination formula step-by-step for [tex]\( n = 250 \)[/tex] and [tex]\( k = 2 \)[/tex]:
1. Factorials:
- [tex]\( 250! \)[/tex] refers to the factorial of 250, which is the product of all positive integers up to 250.
- [tex]\( 2! \)[/tex] refers to the factorial of 2, which is [tex]\( 2 \times 1 = 2 \)[/tex].
- [tex]\( (250 - 2)! = 248! \)[/tex] refers to the factorial of 248, which is the product of all positive integers up to 248.
2. Combination Formula:
[tex]\[
C(250, 2) = \frac{250!}{2! \times 248!}
\][/tex]
3. Simplifying the Formula:
- The factorial [tex]\( 250! \)[/tex] consists of all integers from 1 to 250.
- The factorial [tex]\( 248! \)[/tex] consists of all integers from 1 to 248.
- When dividing [tex]\( 250! \)[/tex] by [tex]\( 248! \)[/tex], many terms cancel out, leaving [tex]\( 250 \times 249 \)[/tex].
Thus, the combination formula simplifies to:
[tex]\[
C(250, 2) = \frac{250 \times 249}{2}
\][/tex]
The correct expression from the provided options is:
[tex]\[
\frac{250!}{(250-2)! \times 2!}
\][/tex]
Putting it all together, the number of possible pairs of volunteers is given by:
[tex]\[
\boxed{\frac{250!}{248! \times 2!}}
\][/tex]
Evaluating it, we get:
[tex]\[
C(250, 2) = 31125
\][/tex]
Thus, the correct choice among the options provided is:
[tex]\[
\frac{250!}{(250-2)! \times 2!}
\][/tex]
