The binomial coefficient $\binom{n}{k}$, also known as "n choose k," is a fundamental concept in combinatorics. It represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to the order of selection. The formula to compute $\binom{n}{k}$ is:

$\binom{n}{k} = \frac{n!}{k!(n-k)!}$

where $n!$ (n factorial) is the product of all positive integers up to $n$, defined as:

$n! = n \times (n-1) \times (n-2) \times \cdots \times 1$

and similarly for $k!$ and $(n-k)!$.

Steps to Compute $\binom{n}{k}$

1. Calculate the factorial of $n$: $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$

2. Calculate the factorial of $k$: $k! = k \times (k-1) \times (k-2) \times \cdots \times 1$

3. Calculate the factorial of $(n-k)$: $(n-k)! = (n-k) \times (n-k-1) \times \cdots \times 1$

4. Substitute these values into the binomial coefficient formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

Example Calculation

Let's compute $\binom{5}{2}$:

1. Calculate $5!$: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

2. Calculate $2!$: $2! = 2 \times 1 = 2$

3. Calculate $(5-2)! = 3!$: $3! = 3 \times 2 \times 1 = 6$

4. Substitute into the formula: $\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$

So, $\binom{5}{2} = 10$.

Simplified Formula for Small Values

For small values of $k$ relative to $n$, it may be more efficient to cancel terms directly rather than computing the full factorials. For example:

$\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10$

This approach avoids computing the entire factorials when possible.

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Related Questions:

1. How do binomial coefficients relate to Pascal's triangle?
2. What are some common properties of binomial coefficients?
3. How is the binomial coefficient used in the binomial theorem?
4. Can you provide a real-world application of binomial coefficients?
5. How do you simplify expressions involving multiple binomial coefficients?
6. How do binomial coefficients appear in combinatorial proofs?
7. What is the relationship between binomial coefficients and combinations?
8. How can you compute binomial coefficients using dynamic programming?

Tip:

When dealing with large factorials, consider using properties of binomial coefficients or computational tools to avoid overflow and improve efficiency.