To determine the probability of randomly drawing the tiles with the letters [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] from a bag containing 26 tiles (each with a different letter of the alphabet), we can follow these steps:
1. Identify the Total Number of Tiles and Chosen Tiles:
The bag contains 26 tiles (one for each letter of the alphabet), and we are choosing 3 tiles.
2. Calculate the Total Number of Possible Combinations:
The total number of ways to choose 3 tiles from 26 tiles can be computed using the combination formula, denoted as [tex]\( \binom{n}{k} \)[/tex]:
[tex]\[
\binom{26}{3} = \frac{26!}{3!(26-3)!} = \frac{26!}{3! \cdot 23!}
\][/tex]
This simplifies to:
[tex]\[
\binom{26}{3} = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = 2600
\][/tex]
3. Count the Number of Favorable Outcomes:
There is only one specific combination of drawing the tiles [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]. Hence, there is only 1 favorable outcome.
4. Calculate the Probability:
The probability of drawing the tiles [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] is given by the ratio of the number of favorable outcomes to the total number of possible combinations:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of combinations}} = \frac{1}{2600}
\][/tex]
So, the probability that you choose the tiles with the letters [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] is [tex]\(\frac{1}{2600}\)[/tex].
The correct answer is:
[tex]\[ \boxed{\frac{1}{2600}} \][/tex]
