Certainly! Let's find out how many unique ways we can select 3 marbles from a bag containing 8 marbles of different colors. To solve this, we will use the concept of combinations.
### Understanding Combinations
Combinations refer to the selection of items without considering the order. The formula to calculate the number of combinations of selecting [tex]\( r \)[/tex] items from [tex]\( n \)[/tex] items is given by:
[tex]\[ C(n, r) = \frac{n!}{r!(n-r)!} \][/tex]
Here,
- [tex]\( n \)[/tex] is the total number of items,
- [tex]\( r \)[/tex] is the number of items to be selected, and
- [tex]\( ! \)[/tex] denotes factorial, which means the product of all positive integers up to that number.
### Applying the Combination Formula
In this question,
- [tex]\( n = 8 \)[/tex] (since there are 8 marbles),
- [tex]\( r = 3 \)[/tex] (since we want to select 3 marbles).
Using the combination formula:
[tex]\[ C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!} \][/tex]
### Simplification
Factorials can be large numbers, so it is often helpful to cancel out common terms in the numerator and the denominator to simplify the calculation.
[tex]\[ C(8, 3) = \frac{8 \times 7 \times 6 \times 5!}{3! \times 5!} \][/tex]
Here, [tex]\( 5! \)[/tex] in the numerator and denominator cancel each other out:
[tex]\[ C(8, 3) = \frac{8 \times 7 \times 6}{3!} \][/tex]
Now, calculate [tex]\( 3! \)[/tex]:
[tex]\[ 3! = 3 \times 2 \times 1 = 6 \][/tex]
Then substitute back in:
[tex]\[ C(8, 3) = \frac{8 \times 7 \times 6}{6} \][/tex]
Finally, the 6 in the numerator and denominator cancel each other out:
[tex]\[ C(8, 3) = 8 \times 7 = 56 \][/tex]
### Conclusion
Therefore, there are [tex]\( \boxed{56} \)[/tex] unique ways to select 3 marbles from a bag of 8 marbles.
