Final answer:
The number of different samples of 3 pieces of candy that can be pulled from a cauldron containing 45 total pieces is calculated using combinations, specifically the formula C(45, 3), which takes into account the total number of items and the number desired for the sample, without considering the order of selection.
Step-by-step explanation:
To determine how many different samples of 3 pieces of candy can be pulled from a cauldron containing 45 total pieces, we use the concept of combinations in mathematics. The formula for a combination when you are choosing 'k' items from 'n' possibilities, without regard to the order of selection, is given by:
C(n, k) = n! / (k!(n - k)!)
Where 'n' is the total number of items to choose from, 'k' is the number of items to choose, and '!' denotes the factorial of a number.
Applying this to our problem:
- There are 45 pieces of candy in total, so 'n' is 45.
- We want to pick 3 pieces, so 'k' is 3.
- The combination formula becomes C(45, 3) = 45! / (3!(45 - 3)!).
- Calculating the factorials and the division yields a result which shows us the total number of different samples possible.
The answer to the question gives the number of combinations possible when selecting 3 pieces of candy from a total of 45 without replacement.