91Ó°ÊÓ

When asked to find how many different three-letter initials begin with an A, we need to break down the problem into smaller parts.
Each initial consists of three letters. The problem specifies that the first letter must be an 'A'.
Since the first letter is fixed, we only have one option for this letter.

Moving on to the second and third letters, we have 26 choices from the English alphabet for each of these positions.
So, for each position, we consider the full range of possibilities: A, B, C, ..., Z.
By isolating and understanding each part of the problem, we can effectively calculate the total number of possible combinations.

Permutations are different ways in which a set of items can be arranged. In this problem, we are dealing with a special case where some positions are fixed.
For example, the first letter is always 'A', which means we are only permuting the remaining two letters.
It is important to remember that permutations consider the order of elements.
Thus, the initials AB and BA are different permutations.

In our scenario, the calculation of permutations begins by fixing the first letter and then computing the possible choices for the second and third letters.
Even though it might seem simple, understanding permutations helps in problems where we need to arrange or sequence different items.

The Basic Counting Principle is a fundamental concept used in combinatorics to find the total number of outcomes for a series of events.
It states that if one event can occur in 'm' ways and a second event can occur independently of the first in 'n' ways, then the two events can occur in ways.

In this problem, we have a three-step event where:

• The first letter is 'A' (1 way)


• The second letter can be any of the 26 letters (26 ways)


• The third letter can also be from the 26 letters (26 ways)

To find the total number of different three-letter initials beginning with 'A', we multiply the number of ways for each step:
This gives us: \[\begin{equation} 1 \times 26 \times 26 = 676 \end{equation}\] Hence, by applying the Basic Counting Principle, we determine that there are 676 different three-letter initials starting with 'A'.