Certainly! Let's go through the problem step-by-step.
### Step 1: Understanding Notation
In this expression, [tex]\(\binom{n}{k}\)[/tex] represents a binomial coefficient, which is the number of ways to choose [tex]\(k\)[/tex] items from [tex]\(n\)[/tex] items without regard to order. It is calculated using the formula:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Where [tex]\(n!\)[/tex] denotes the factorial of [tex]\(n\)[/tex].
### Step 2: Compute [tex]\(\binom{8}{4}\)[/tex]
Let's first calculate [tex]\(\binom{8}{4}\)[/tex]:
[tex]\[
\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!}
\][/tex]
The value of [tex]\(8!\)[/tex] (8 factorial) is:
[tex]\[
8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320
\][/tex]
The value of [tex]\(4!\)[/tex] (4 factorial) is:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
So, we have:
[tex]\[
\binom{8}{4} = \frac{40320}{24 \times 24} = \frac{40320}{576} = 70
\][/tex]
### Step 3: Compute [tex]\(\binom{11}{4}\)[/tex]
Next, let's compute [tex]\(\binom{11}{4}\)[/tex]:
[tex]\[
\binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11!}{4!7!}
\][/tex]
The value of [tex]\(11!\)[/tex] (11 factorial) is:
[tex]\[
11! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 39916800
\][/tex]
We already know that [tex]\(4! = 24\)[/tex], and the value of [tex]\(7!\)[/tex] (7 factorial) is:
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\][/tex]
So, we have:
[tex]\[
\binom{11}{4} = \frac{39916800}{24 \times 5040} = \frac{39916800}{120960} = 330
\][/tex]
### Step 4: Perform the Division
Now that we have [tex]\(\binom{8}{4} = 70\)[/tex] and [tex]\(\binom{11}{4} = 330\)[/tex], let's perform the indicated division [tex]\(\frac{\binom{8}{4}}{\binom{11}{4}}\)[/tex]:
[tex]\[
\frac{\binom{8}{4}}{\binom{11}{4}} = \frac{70}{330} = \frac{7}{33} \approx 0.21212121212121213
\][/tex]
### Conclusion
Thus, the step-by-step solution to find the ratio [tex]\(\frac{\binom{8}{4}}{\binom{11}{4}}\)[/tex] results in approximately [tex]\(0.2121\)[/tex], or more precisely:
[tex]\[
\frac{70}{330} \approx 0.21212121212121213
\][/tex]
So, the result of the division is [tex]\(0.21212121212121213\)[/tex].
