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.