The expression \( \frac{n!}{r!} \) is a ratio involving factorials, which have many applications in combinatorics and probability.

### 1. **Understanding Factorials**:
   - **Factorial (n!)**: The factorial of a number \(n\), denoted \(n!\), is the product of all positive integers from 1 to \(n\). Mathematically,
     \[
     n! = n \times (n-1) \times (n-2) \times \cdots \times 1
     \]
     For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

   - **Factorial of \(r\) (r!)**: Similarly, \(r!\) is the product of all integers from 1 to \(r\):
     \[
     r! = r \times (r-1) \times \cdots \times 1
     \]
   
### 2. **The Ratio \(\frac{n!}{r!}\)**:
   The ratio of two factorials \( \frac{n!}{r!} \) simplifies the product of integers by canceling terms. Since \( r! \) is a subset of \( n! \) for \( r \leq n \), you can rewrite the ratio as:
   \[
   \frac{n!}{r!} = \frac{n \times (n-1) \times \cdots \times (r+1) \times r!}{r!}
   \]
   Cancelling \( r! \) from the numerator and denominator gives:
   \[
   \frac{n!}{r!} = n \times (n-1) \times \cdots \times (r+1)
   \]
   This result represents the product of integers from \(n\) down to \(r+1\), excluding all terms in \(r!\).

### 3. **Applications**:
   - **Permutations**: This ratio often arises when computing permutations. The number of ways to arrange \(n\) objects, taking \(r\) at a time, is given by:
     \[
     P(n, r) = \frac{n!}{(n-r)!}
     \]
   - **Combinations**: In combinations, where order doesn't matter, this ratio is used to count the ways to choose \(r\) objects from \(n\), and is represented as:
     \[
     C(n, r) = \frac{n!}{r!(n-r)!}
     \]

### 4. **Key Properties**:
   - **Cancelling Factorials**: Since factorials grow very fast, simplifying expressions like \( \frac{n!}{r!} \) helps reduce the computation to a manageable product.
   - **Usage in Binomial Coefficients**: This ratio also shows up in the binomial theorem, where the coefficient of \(x^r\) in \((1 + x)^n\) is \( \frac{n!}{r!(n-r)!} \).

### Example:
Let’s compute \( \frac{7!}{4!} \):
\[
7! = 7 \times 6 \times 5 \times 4!
\]
\[
\frac{7!}{4!} = \frac{7 \times 6 \times 5 \times 4!}{4!} = 7 \times 6 \times 5 = 210
\]

This gives us \( \frac{7!}{4!} = 210 \).

Would you like to dive deeper into any specific aspect, such as applications of this ratio or working through more examples?

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### Related Questions:
1. What are the common uses of factorials in probability theory?
2. How does the formula for permutations differ from combinations?
3. How do factorials help in solving recurrence relations?
4. What are some interesting patterns in factorial growth rates?
5. Can you explain the binomial theorem in terms of factorials?

### Tip:
When simplifying ratios of factorials, always look for terms that cancel out to make calculations easier!

Explore the mathematical concept of factorials and how the ratio n!/r! simplifies in combinatorics. Learn key applications such as permutations, combinations, and the binomial theorem. Step-by-step examples provided, suitable for students in Grades 9-12.

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