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!