Sure, let's break this problem down step-by-step.
1. Understanding the Problem:
- We have a bag containing 26 tiles, each one representing a different letter from A to Z.
- We need to determine the probability that when we choose 3 tiles from the bag, they will specifically be the letters A, B, and C.
2. Total Number of Selection Outcomes:
- First, we need to find the total number of ways to choose 3 tiles out of the 26 tiles.
- This can be calculated using combinations (also known as "binomial coefficients"), which is denoted as C(n, k) where n is the total number of items to choose from, and k is the number of items to choose.
- Mathematically, it’s written as [tex]\( C(26, 3) \)[/tex].
3. Formula for Combinations:
- The formula for combinations is:
[tex]\[
C(n, k) = \frac{n!}{k!(n-k)!}
\][/tex]
- Plug in the values for our problem:
[tex]\[
C(26, 3) = \frac{26!}{3!(26-3)!} = \frac{26!}{3! \times 23!} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600
\][/tex]
- Therefore, there are 2600 different ways to choose 3 tiles from a set of 26 tiles.
4. Successful Outcomes:
- We are interested in the specific outcome where we choose the tiles A, B, and C.
- There is only one way to choose this specific combination of A, B, and C out of the 26 tiles.
- So, the number of favorable outcomes is 1.
5. Calculating the Probability:
- The probability of an event is given by the number of favorable outcomes divided by the total number of possible outcomes.
- Therefore, the probability [tex]\( P \)[/tex] is:
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{2600}
\][/tex]
6. Conclusion:
- The probability that the 3 tiles chosen from the bag consist of the letters A, B, and C is [tex]\( \frac{1}{2600} \)[/tex].
That completes our step-by-step solution. So, the probability that the tiles you chose consist of the letters A, B, and C is [tex]\( \frac{1}{2600} \)[/tex].
