If 2 letters are chosen from the English alphabet, then the probability of both the letters being vowels is :

Question

Accepted Answer

To determine the probability that both letters chosen from the English alphabet are vowels, we need to follow these steps:

1. **Identify the Total Number of Letters and Vowels:**
   - The English alphabet consists of 26 letters.
   - There are 5 vowels in the English alphabet: A, E, I, O, and U.

2. **Calculate the Total Number of Ways to Choose 2 Letters:**
   - The total number of ways to choose 2 letters from 26 is given by the combination formula $ \binom{n}{r} $, where $ n $ is the total number of items to choose from, and $ r $ is the number of items to choose.
   - Therefore, the total number of ways to choose 2 letters from 26 is $ \binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325 $.

3. **Calculate the Number of Ways to Choose 2 Vowels:**
   - The number of ways to choose 2 vowels from the 5 available vowels is $ \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 $.

4. **Calculate the Probability:**
   - The probability that both letters chosen are vowels is the number of favorable outcomes divided by the total number of possible outcomes.
   - Therefore, the probability is $ \frac{10}{325} = \frac{2}{65} $.

5. **Explanation of Why Option 2 is Correct:**
   - The calculation shows that the probability of both letters being vowels is $ \frac{2}{65} $, which matches Option 2. This confirms that Option 2 is the correct answer.

6. **Explanation of Why Other Options are Incorrect:**
   - **Option 1: $\frac{1}{65}$**: This option is incorrect because the calculated probability is $\frac{2}{65}$, not $\frac{1}{65}$.
   - **Option 3: $\frac{3}{65}$**: This option is incorrect because the calculated probability is $\frac{2}{65}$, not $\frac{3}{65}$.
   - **Option 4: $\frac{4}{65}$**: This option is incorrect because the calculated probability is $\frac{2}{65}$, not $\frac{4}{65}$.

By following the logical steps and calculations above, we have demonstrated that Option 2 is indeed the correct answer, as it accurately represents the probability of both letters being vowels when two letters are chosen from the English alphabet.