To solve this problem, we need to count the number of 3-element subsets from the set $\{A, B, C, D, E, F, G\}$ that include the letter $A$.

Step-by-Step Solution:

1. Since the subset must include $A$, we select $A$ as one of the elements, leaving us to choose the remaining two elements from the set $\{B, C, D, E, F, G\}$.

2. The set $\{B, C, D, E, F, G\}$ has 6 elements, and we need to choose 2 of them.

3. The number of ways to choose 2 elements from a set of 6 is given by the combination formula: $\binom{6}{2} = \frac{6!}{2!(6 - 2)!} = \frac{6 \times 5}{2 \times 1} = 15$

Answer:

There are 15 different 3-element subsets containing the letter $A$ that can be formed from the set $\{A, B, C, D, E, F, G\}$.

Would you like further details or have any questions?

Here are some related questions to consider:

1. How many 3-element subsets can be formed without any restrictions on specific elements?
2. How many subsets of size 4 containing the letter $B$ can be formed from the set $\{A, B, C, D, E, F, G\}$?
3. How many 4-element subsets can be created from the set $\{A, B, C, D, E, F, G\}$?
4. How many 3-element subsets containing at least one of $B$ or $C$ can be formed?
5. How would the answer change if we were forming 4-element subsets containing the letter $A$?

Tip: Remember, when dealing with subsets that must contain specific elements, it’s often helpful to fix those elements first and then calculate combinations for the remaining choices.