Three letters are randomly selected from the 26 capital letters of the english alphabet. What is the probability that the letter 'A' will not be included in the choice?

Options

A. `1//2`
B. `23//26`
C. `12//13`
D. `25//26`

Answer

To find the probability that the letter 'A' will not be included when selecting three letters from the 26 capital letters of the English alphabet, we can follow these steps:

### Step 1: Determine the total number of ways to choose 3 letters from 26 letters.
The total number of ways to choose 3 letters from 26 is given by the combination formula \( C(n, r) \), which is calculated as:

\[ C(26, 3) = \frac{26!}{3!(26-3)!} = \frac{26!}{3! \cdot 23!} \]

### Step 2: Calculate \( C(26, 3) \).
Calculating \( C(26, 3) \):

\[ C(26, 3) = \frac{26 \times 25 \times 24}{3 \times 2 \times 1} = \frac{15600}{6} = 2600 \]

### Step 3: Determine the number of ways to choose 3 letters excluding 'A'.
If 'A' is not included, we are left with 25 letters (B to Z). The number of ways to choose 3 letters from these 25 letters is:

\[ C(25, 3) = \frac{25!}{3!(25-3)!} = \frac{25!}{3! \cdot 22!} \]

### Step 4: Calculate \( C(25, 3) \).
Calculating \( C(25, 3) \):

\[ C(25, 3) = \frac{25 \times 24 \times 23}{3 \times 2 \times 1} = \frac{13800}{6} = 2300 \]

### Step 5: Calculate the probability that 'A' is not included.
The probability that 'A' is not included in the selection of 3 letters is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(\text{'A' not included}) = \frac{C(25, 3)}{C(26, 3)} = \frac{2300}{2600} \]

### Step 6: Simplify the probability.
Now, simplifying \( \frac{2300}{2600} \):

\[ P(\text{'A' not included}) = \frac{2300 \div 300}{2600 \div 300} = \frac{23}{26} \]

### Final Answer:
Thus, the probability that the letter 'A' will not be included in the choice is:

\[ \frac{23}{26} \]