Certainly! Let's break down the problem and find the number of unique ways to select 3 marbles from a bag containing 8 marbles of different colors.
### Step-by-Step Solution:
1. Understand the problem:
We need to find the number of ways to choose 3 marbles out of 8 without considering the order. This is a problem of combinations.
2. Definition of combinations:
The number of ways to choose [tex]\(k\)[/tex] items from [tex]\(n\)[/tex] items without regard to order is given by the combination formula:
[tex]\[
C(n, k) = \frac{n!}{k!(n-k)!}
\][/tex]
where [tex]\(n!\)[/tex] (n factorial) is the product of all positive integers up to [tex]\(n\)[/tex].
3. Apply the formula:
Here, we have [tex]\(n = 8\)[/tex] and [tex]\(k = 3\)[/tex]. Substituting these values into the combination formula gives:
[tex]\[
C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!}
\][/tex]
Note: The factorial function [tex]\(n!\)[/tex] simplifies the calculation by canceling parts of the original factorial terms.
4. Simplify the calculation:
Let's break down the factorials:
- [tex]\(8! = 8 \times 7 \times 6 \times 5!\)[/tex]
- [tex]\(3! = 3 \times 2 \times 1 = 6\)[/tex]
- [tex]\(5!\)[/tex] is a common term in both the numerator and denominator and cancels out.
Therefore,
[tex]\[
\frac{8!}{3! \cdot 5!} = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} = \frac{8 \times 7 \times 6}{3!} = \frac{8 \times 7 \times 6}{6}
\][/tex]
5. Calculate the result:
Now simplify the fraction by performing the division:
[tex]\[
\frac{8 \times 7 \times 6}{6} = 8 \times 7 = 56
\][/tex]
So, the number of unique ways to select 3 marbles from 8 marbles is [tex]\( \boxed{56} \)[/tex].
