To evaluate the expression [tex]\(\frac{15!}{6!}\)[/tex], let's break it down step-by-step:
1. Factorial Calculation:
- The factorial of a number [tex]\(n\)[/tex], denoted as [tex]\(n!\)[/tex], is the product of all positive integers less than or equal to [tex]\(n\)[/tex].
- So, [tex]\(15!\)[/tex] (15 factorial) and [tex]\(6!\)[/tex] (6 factorial) will need to be computed.
2. Calculate [tex]\(15!\)[/tex]:
- [tex]\(15! = 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\)[/tex].
- After calculating, [tex]\(15! = 1,307,674,368,000\)[/tex].
3. Calculate [tex]\(6!\)[/tex]:
- [tex]\(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\)[/tex].
- After calculating, [tex]\(6! = 720\)[/tex].
4. Divide the Factorials:
- Now we find [tex]\(\frac{15!}{6!}\)[/tex] by dividing the factorial of 15 by the factorial of 6.
- [tex]\(\frac{1,307,674,368,000}{720} = 1,816,214,400.0\)[/tex].
Therefore, the simplified answer is:
[tex]\[
\frac{15!}{6!} = 1,816,214,400.0
\][/tex]
