Fill in the blank so that the resulting statement is true.
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Part 1
The number of possible combinations if r objects are taken from n items is Subscript n Baseline Upper C Subscript requals
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n exclamation mark .
StartFraction n exclamation mark Over left parenthesis n minus r right parenthesis exclamation mark r exclamation mark EndFraction .
StartFraction n exclamation mark Over r exclamation mark EndFraction .
StartFraction n exclamation mark Over left parenthesis n minus r right parenthesis exclamation mark EndFraction .

To fill in the blank with the correct formula for the number of possible combinations if \( r \) objects are taken from \( n \) items, the correct expression is:

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

So the complete statement should be:

"The number of possible combinations if \( r \) objects are taken from \( n \) items is \(\binom{n}{r} = \frac{n!}{(n-r)!r!}\)."

The correct option to fill in the blank is:

**StartFraction n exclamation mark Over left parenthesis n minus r right parenthesis exclamation mark r exclamation mark EndFraction .**

Would you like more details on combinations, or do you have any questions?

Here are some related questions:
1. How do you calculate the number of combinations for specific values of \( n \) and \( r \)?
2. What is the difference between permutations and combinations?
3. How does the formula for combinations change if \( r = 0 \)?
4. Can the formula for combinations be derived using factorials?
5. What is the relationship between combinations and Pascal's Triangle?

**Tip:** Remember, combinations are used when the order of selection doesn't matter, while permutations are for when the order does matter.

Learn how to calculate the number of possible combinations if \( r \) objects are taken from \( n \) items using the combination formula. This problem covers fundamental concepts in combinatorics and factorial calculations. Suitable for high school students in Grades 10-12.

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