To determine the number of possible combinations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time, we use the combination formula, represented as:
[tex]\[
C(n, r) = \frac{n!}{r!(n-r)!}
\][/tex]
Here's a step-by-step explanation:
1. Understanding the Concepts:
- A combination is a way of selecting items from a larger pool, where the order of selection does not matter.
- The notation [tex]\( C(n, r) \)[/tex] represents the number of combinations of [tex]\( n \)[/tex] items taken [tex]\( r \)[/tex] at a time.
2. Formula Explanation:
- [tex]\( n! \)[/tex] denotes the factorial of [tex]\( n \)[/tex], which is the product of all positive integers up to [tex]\( n \)[/tex].
- [tex]\( r! \)[/tex] denotes the factorial of [tex]\( r \)[/tex].
- [tex]\( (n-r)! \)[/tex] represents the factorial of [tex]\( n-r \)[/tex].
3. Using the Formula:
- The expression [tex]\( C(n, r) = \frac{n!}{r!(n-r)!} \)[/tex] incorporates all the necessary elements to calculate the number of combinations.
Now let's evaluate the given expressions to see which one matches the formula for combinations:
1. Expression: [tex]\(\frac{n \mid}{(n-r)|r|}\)[/tex]
- This expression does not match any standard mathematical notation and seems incorrect.
2. Expression: [tex]\(\frac{n!}{(n-r)!}\)[/tex]
- This is the formula for permutations, [tex]\( P(n, r) \)[/tex], and not for combinations. In permutations, the order matters, which is different from combinations.
3. Expression: [tex]\(\frac{(n-r)!}{n!}\)[/tex]
- This expression is inverted compared to the proper formula for combinations and does not consider the requirement of dividing by [tex]\( r! \)[/tex].
4. Expression: [tex]\(\frac{(n-r)!r!}{n!}\)[/tex]
- This expression also does not match the combination formula. It seems to have the factorials in the wrong places.
Therefore, none of the provided options is directly matching the correct combination formula [tex]\( \frac{n!}{r!(n-r)!} \)[/tex].
Since the correct formulation [tex]\( \frac{n!}{r!(n-r)!} \)[/tex] is not in the options, it might indicate an error in the options provided, or potentially a misrepresentation of one of the options. Based on the classical combination formula, the correct expression should be recognized as none of the above if strictly adhering to the correct form.
