The expression you provided is:

$\frac{38!}{(38-21)!21!} = \frac{38!}{17! \cdot 21!}$

This is a combination formula, often written as:

$C(38, 21) = \frac{38!}{(38-21)!21!} = \frac{38!}{17!21!}$

Now, let's solve for $C(38, 21)$.

First, we use the formula:

$C(n, r) = \frac{n!}{(n-r)!r!}$

Thus,

$C(38, 21) = \frac{38 \times 37 \times \cdots \times 18}{21!}$

Let me calculate the result for you.The value of $\frac{38!}{17! \cdot 21!}$ or $C(38, 21)$ is $28,781,143,380$.

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How is the combination formula derived?
2. What is the difference between combinations and permutations?
3. How would the answer change if the numbers were $C(38, 20)$?
4. Can you calculate the value of $C(40, 22)$?
5. How do combinations apply in probability theory?

Tip: Combinations are used when the order of selection does not matter, while permutations are used when the order does matter.