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`

Explanation

To solve the problem of finding the probability that the letter 'A' will not be included when selecting 3 letters from the 26 capital letters of the English alphabet, we can follow these steps:

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

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

### Step 2: Determine the number of ways to select 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:

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

### Step 3: Calculate the probability that 'A' is not included.
The probability \( P \) that 'A' will not be included in the selection of 3 letters is the ratio of the number of favorable outcomes to the total outcomes:

\[ P(\text{not A}) = \frac{\text{Favorable ways}}{\text{Total ways}} = \frac{C(25, 3)}{C(26, 3)} \]

Substituting the values we calculated:

\[ P(\text{not A}) = \frac{\frac{25!}{3! \cdot 22!}}{\frac{26!}{3! \cdot 23!}} \]

### Step 4: Simplify the expression.
We can simplify this expression by canceling out the common factorials:

\[ P(\text{not A}) = \frac{25! \cdot 23!}{26! \cdot 22!} \]

Now, we can express \( 26! \) as \( 26 \cdot 25! \):

\[ P(\text{not A}) = \frac{25! \cdot 23!}{26 \cdot 25! \cdot 22!} \]

Canceling \( 25! \) gives:

\[ P(\text{not A}) = \frac{23!}{26 \cdot 22!} \]

Now, we can express \( 23! \) as \( 23 \cdot 22! \):

\[ P(\text{not A}) = \frac{23 \cdot 22!}{26 \cdot 22!} \]

Canceling \( 22! \) gives:

\[ P(\text{not A}) = \frac{23}{26} \]

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

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