Fill in the blank so that the resulting statement is true.

The number of possible permutations if r objects are taken from n items is 

Subscript n Baseline Upper P Subscript rnPrequals=​_____.

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Part 1

The number of possible permutations if r objects are taken from n items is 

Subscript n Baseline Upper P Subscript rnPrequals=

▼
 
StartFraction r exclamation mark Over left parenthesis n minus r right parenthesis exclamation mark EndFraction .r!(n−r)!.

StartFraction n exclamation mark Over left parenthesis r minus n right parenthesis exclamation mark EndFraction .n!(r−n)!.

StartFraction r exclamation mark Over left parenthesis r minus n right parenthesis exclamation mark EndFraction .r!(r−n)!.

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

The number of possible permutations of \( r \) objects taken from \( n \) items is denoted by \( _nP_r \). The correct formula for this is:

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

So, the correct answer to fill in the blank is:

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

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Would you like further clarification on permutations, or do you have any other questions?

### Related Questions:
1. How does the formula for combinations differ from the formula for permutations?
2. What is the value of \( _5P_3 \) using the permutation formula?
3. How many ways can 4 people be arranged in a line if there are 6 people to choose from?
4. What is the difference between permutations and combinations?
5. How do you calculate the number of permutations when some objects are identical?

**Tip:** Remember, in permutations, the order of objects matters, while in combinations, it does not.

Explore permutations in mathematics with the nPr formula explained. Learn how to calculate the number of permutations when r objects are taken from n items. Suitable for high school students, this page covers permutations, factorials, and mathematical problem-solving techniques.

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