To solve this problem, we need to break down the process of determining how many different sequences of seven digits are possible when telephone numbers are constructed in the described format.

Key Points:

1. First three letters: These are associated with the digits from 2 to 9 on the keypad, and each digit (from 2 to 9) has three possible letters. For example:

   • 2 corresponds to A, B, C
   • 3 corresponds to D, E, F
   • ...
   • 9 corresponds to W, X, Y Therefore, for each of the first three digits, there are 3 possible choices for each letter.

2. Next four digits: These are regular digits from 0 to 9, so for each of the four digits, there are 10 possible choices.

Calculation:

• For the first three letters, each letter can correspond to one of three possible values for the numbers 2 to 9. There are 8 numbers (from 2 to 9), and each of them has 3 possible letter combinations. So, for each of the three positions, there are 8 possible choices for the number, and 3 possible letters associated with each number. Hence, for each of the three positions, there are $8 \times 3 = 24$ possible choices.

• For the last four digits, there are 10 choices (0 through 9) for each of the four positions.

So the total number of possible sequences is:

$24^3 \times 10^4 = 24 \times 24 \times 24 \times 10 \times 10 \times 10 \times 10$

Let's calculate that:

$24^3 = 24 \times 24 = 576, \quad 576 \times 24 = 13,824$ $10^4 = 10 \times 10 \times 10 \times 10 = 10,000$ $13,824 \times 10,000 = 138,240,000$

Thus, the total number of different sequences of seven digits possible is 138,240,000.

Do you have any questions or want further details?

Here are 5 related questions to deepen your understanding:

1. How would the number of sequences change if letters were not limited to the digits 2 to 9?
2. What if one of the numbers could not be used (e.g., 5 is removed from the digit-letter association)?
3. How many different sequences would there be if only two letters were associated with each digit?
4. How does the size of the keypad (digits 0-9) impact the total number of possible sequences?
5. How would adding more digits to the sequence affect the total possibilities?

Tip: When dealing with problems involving choices and sequences, remember to multiply the number of choices for each position to find the total number of possibilities.