To evaluate [tex]\(\frac{24!}{21!}\)[/tex], let's break it down step-by-step:
1. Understanding Factorials:
- The factorial of a non-negative integer [tex]\( n \)[/tex] (denoted [tex]\( n! \)[/tex]) is the product of all positive integers up to [tex]\( n \)[/tex].
- For example, [tex]\( 24! \)[/tex] means multiplying all the numbers from 1 to 24 together:
[tex]\[
24! = 24 \times 23 \times 22 \times \ldots \times 2 \times 1
\][/tex]
- Similarly, [tex]\( 21! \)[/tex] means multiplying all the numbers from 1 to 21 together:
[tex]\[
21! = 21 \times 20 \times 19 \times \ldots \times 2 \times 1
\][/tex]
2. Simplifying the Expression:
- Notice that [tex]\( 24! \)[/tex] can be expressed as:
[tex]\[
24! = 24 \times 23 \times 22 \times 21!
\][/tex]
- This allows us to cancel out the [tex]\( 21! \)[/tex] in the numerator and the denominator:
[tex]\[
\frac{24!}{21!} = \frac{24 \times 23 \times 22 \times 21!}{21!}
\][/tex]
- After canceling [tex]\( 21! \)[/tex] from both the numerator and the denominator:
[tex]\[
\frac{24!}{21!} = 24 \times 23 \times 22
\][/tex]
3. Calculating the Product:
- Now we need to multiply these three numbers together:
[tex]\[
24 \times 23 = 552
\][/tex]
[tex]\[
552 \times 22 = 12144
\][/tex]
Thus, the value of [tex]\(\frac{24!}{21!}\)[/tex] is [tex]\( 12144 \)[/tex].
### Summary:
[tex]\[
\frac{24!}{21!} = 24 \times 23 \times 22 = 12144
\][/tex]
