The combination formula, often denoted as \( \binom{n}{k} \) (read as "n choose k"), represents the number of ways to choose \( k \) items from a set of \( n \) items without regard to the order in which they are chosen.

### Combination Formula
The formula for combinations is:

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

where:
- \( n! \) (n factorial) is the product of all positive integers from 1 to \( n \) (i.e., \( n! = n \times (n-1) \times \cdots \times 2 \times 1 \)).
- \( k! \) (k factorial) is the product of all positive integers from 1 to \( k \).
- \( (n-k)! \) is the factorial of \( (n - k) \).

### Explanation of the Formula
- The numerator \( n! \) counts all the possible ways to arrange \( n \) items.
- The denominator \( k! \) counts the arrangements of the \( k \) items selected, and \( (n-k)! \) counts the arrangements of the items not selected. Since the order does not matter in combinations, we divide by these to avoid overcounting.

### Example Calculation

Suppose we want to calculate \( \binom{5}{2} \), the number of ways to choose 2 items from a set of 5:

\[
\binom{5}{2} = \frac{5!}{2! \cdot (5-2)!} = \frac{5!}{2! \cdot 3!}
\]

Calculating the factorials:

- \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
- \( 2! = 2 \times 1 = 2 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

Plugging these into the formula:

\[
\binom{5}{2} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10
\]

So, there are 10 ways to choose 2 items from a set of 5.

Would you like further clarification, or do you have any questions?

### 5 Related Questions:
1. How would you calculate \( \binom{n}{3} \) for any integer \( n \)?
2. What is the value of \( \binom{7}{4} \), and how does it relate to \( \binom{7}{3} \)?
3. If \( \binom{n}{2} = 45 \), what is the value of \( n \)?
4. How does the combination formula change if repetition of items is allowed?
5. Can the combination formula be used to calculate permutations? If not, what is the difference?

**Tip:** The combination formula is particularly useful in probability and statistics when determining the likelihood of various outcomes.

Explore the combination formula \( \binom{n}{k} \) and its applications. Learn how to calculate combinations using factorial notation. Example calculations included.

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