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

The formula for 

Subscript n Baseline Upper C Subscript rnCr 

has the same numerator as the formula for 

Subscript n Baseline Upper P Subscript rnPr 

but contains an extra factor of​ _____ in the denominator.

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Part 1

The formula for 

Subscript n Baseline Upper C Subscript rnCr 

has the same numerator as the formula for 

Subscript n Baseline Upper P Subscript rnPr 

but contains an extra factor of 

▼
 
n exclamation markn!

rr

left parenthesis n minus r right parenthesis exclamation mark(n−r)!

r exclamation markr!
 
in the denominator.

To fill in the blank, let's compare the formulas for combinations (\( nCr \)) and permutations (\( nPr \)):

1. **Permutations formula**:
   \[
   nPr = \frac{n!}{(n-r)!}
   \]

2. **Combinations formula**:
   \[
   nCr = \frac{n!}{(n-r)! \cdot r!}
   \]

As we can see, both formulas share the same numerator (\( n! \)). However, the combinations formula has an extra factor of \( r! \) in the denominator.

So the complete statement should be:

"The formula for \( nCr \) has the same numerator as the formula for \( nPr \) but contains an extra factor of \( r! \) in the denominator."

The correct option to fill in the blank is:

**\( r! \)**

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

Here are some related questions:
1. How does the concept of permutations differ from combinations in practical applications?
2. What is the significance of the factorial function in counting problems?
3. How can you derive the permutation formula from the combination formula?
4. Can the combination formula be used to find permutations? If so, how?
5. What are some examples of problems where you would use combinations instead of permutations?

**Tip:** When dealing with combinations, the order of selection doesn't matter, so we divide by \( r! \) to account for the different possible orders.

Explore the differences between combinations (nCr) and permutations (nPr) in mathematics. Understand how the factorial function influences these formulas and their applications. Suitable for high school students and anyone interested in combinatorial mathematics.

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