How many combinations are there of 4 out of 10 objects? (Enter answer as a whole number)

To calculate the number of combinations of 4 objects out of 10, we use the combination formula:

\[
C(n, k) = \frac{n!}{k!(n - k)!}
\]

Where:
- \(n = 10\) (total number of objects),
- \(k = 4\) (number of objects to choose).

Now, let's compute it:

\[
C(10, 4) = \frac{10!}{4!(10 - 4)!} = \frac{10!}{4!6!}
\]

Simplifying:

\[
C(10, 4) = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210
\]

So, the number of combinations is \( \boxed{210} \).

Would you like more details or have any questions? 

Here are 5 related questions:
1. How many combinations are there of 5 out of 12 objects?
2. What is the difference between permutations and combinations?
3. If order matters, how would you calculate the number of permutations of 4 out of 10 objects?
4. How do you generalize this formula to choosing \(k\) objects from \(n\) objects?
5. Can combinations be applied to real-life problems like lotteries? How?

**Tip:** When order doesn't matter in selecting objects, always use combinations; otherwise, use permutations!

Learn how to calculate the number of combinations of 4 objects chosen from a set of 10. This example walks through the combination formula and applies it step-by-step, helping you understand fundamental concepts in combinatorics. Ideal for high school students in Grades 9-12.

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