To evaluate the factorial expression [tex]\(\frac{14!}{11!}\)[/tex], let's break it down step by step.
1. Understand Factorials:
- The factorial of a non-negative integer [tex]\( n \)[/tex], denoted as [tex]\( n! \)[/tex], is the product of all positive integers less than or equal to [tex]\( n \)[/tex]. For example:
[tex]\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
2. Write the Expression:
- We need to evaluate [tex]\(\frac{14!}{11!}\)[/tex]. First, let's expand both factorials:
[tex]\[
14! = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
[tex]\[
11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
3. Simplify the Expression:
- Notice that [tex]\(11!\)[/tex] is a common factor in both [tex]\(14!\)[/tex] and [tex]\(11!\)[/tex]. Therefore, we can cancel out [tex]\(11!\)[/tex] in the numerator and the denominator:
[tex]\[
\frac{14!}{11!} = \frac{14 \times 13 \times 12 \times \cancel{11!}}{\cancel{11!}} = 14 \times 13 \times 12
\][/tex]
4. Perform the Multiplication:
- Now we just need to multiply the remaining numbers in the numerator:
[tex]\[
14 \times 13 = 182
\][/tex]
[tex]\[
182 \times 12 = 2184
\][/tex]
So, the value of the expression [tex]\(\frac{14!}{11!}\)[/tex] is:
[tex]\[
2184
\][/tex]
Thus, [tex]\(\frac{14!}{11!} = 2184\)[/tex].
