Problem

Evaluate the following expression.
\[
\frac{10 !}{4 ! 6 !}
\]

Answer

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Answer

Final Answer: The value of the expression \(\frac{10 !}{4 ! 6 !}\) is \(\boxed{210}\).

Steps

Step 1 :The given expression is \(\frac{10 !}{4 ! 6 !}\). This is a combination formula, often used in probability theory and combinatorics. It represents the number of ways to choose 6 items from 10, without considering the order of selection.

Step 2 :The formula for combinations is \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial, which is the product of all positive integers up to that number.

Step 3 :In this case, n = 10 and k = 6. So we need to calculate the value of \(\frac{10!}{4!*6!}\).

Step 4 :After calculating, we find that the result is 210.0.

Step 5 :Final Answer: The value of the expression \(\frac{10 !}{4 ! 6 !}\) is \(\boxed{210}\).

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