Understanding the Context

$\binom{8}{3}$ represents the number of combinations of 8 items taken 3 at a time. It answers the question: In how many different ways can 3 items be selected from a group of 8 unique items?

For example, if you’re selecting a team of 3 players from 8 candidates, $\binom{8}{3} = 56$ means there are 56 distinct combinations possible. This concept is essential in fields like probability, statistics, genetics, computer science, and project planning—where selection without replacement matters.

The Formula: $\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$

The binomial coefficient is formally defined as:

Key Insights

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

Where:
- $n!$ (n factorial) is the product of all positive integers up to $n$:
$n! = n \ imes (n-1) \ imes (n-2) \ imes \cdots \ imes 1$
- $k!$ is the factorial of $k$, and
- $(n - k)!$ is the factorial of the difference.

Plugging in $n = 8$ and $k = 3$:

$$
\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!}
$$

Step-by-Step Calculation

Final Thoughts

To better understand, let's break down the calculation step-by-step:

1. Write out $8!$:
   $8! = 8 \ imes 7 \ imes 6 \ imes 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1$

2. Write out $5!$ (since $8 - 3 = 5$):
   $5! = 5 \ imes 4 \ imes 3 \ imes 2 \ imes 1 = 120$

3. Write out $3!$:
   $3! = 3 \ imes 2 \ imes 1 = 6$

4. Substitute into the formula:
   $$
   \binom{8}{3} = \frac{8 \ imes 7 \ imes 6 \ imes 5!}{3! \ imes 5!}
   $$

5. Cancel $5!$ in numerator and denominator:
   $\binom{8}{3} = \frac{8 \ imes 7 \ imes 6}{3!} = \frac{8 \ imes 7 \ imes 6}{3 \ imes 2 \ imes 1}$