To evaluate \( X(18, 7) = \frac{18!}{7! \cdot 11!} \), we can use the formula for combinations:

\[
X(n, k) = \binom{n}{k} = \frac{n!}{k! \cdot (n-k)!}
\]

Here, \( n = 18 \) and \( k = 7 \), so:

\[
X(18, 7) = \frac{18!}{7! \cdot (18 - 7)!} = \frac{18!}{7! \cdot 11!}
\]

Now, let's compute the value:

1. First, expand the factorial terms where necessary. Notice that \( 18! = 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11! \). This way, \( 11! \) in the numerator and denominator will cancel out:

\[
\frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11!}{7! \cdot 11!} = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12}{7!}
\]

2. Now calculate \( 7! \):

\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\]

3. Substitute the values back into the expression:

\[
\frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12}{5040}
\]

4. Calculate the product of the numerator:

\[
18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 = 183783600
\]

5. Finally, divide by \( 5040 \):

\[
\frac{183783600}{5040} = 364140
\]

Therefore, the value of \( X(18, 7) = \frac{18!}{7! \cdot 11!} \) is **364,140**.

Would you like more details or have any questions?

Here are some related questions:

1. What is the significance of combinations and how are they used in probability?
2. How do you calculate permutations, and what is the difference between permutations and combinations?
3. How does the combination formula change when \( n = k \)?
4. Can you explain the concept of factorials and their properties?
5. How do you simplify expressions involving combinations?

**Tip:** When simplifying factorial expressions, cancel out common terms from the numerator and denominator early to make calculations easier.

Learn how to calculate the combination X(18, 7) = 18! / (7! * 11!) with a step-by-step solution. This problem involves using the combination formula and factorial calculations. Suitable for high school students.

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