How many different groups of 3 students can be chosen from a class of 20, avoiding consecutive names?

To determine the number of ways to select 3 students from a class of 20 while ensuring that no two chosen students have consecutive names, a systematic approach can be applied.

First, label the students from 1 to 20. If you choose a student, the next student selected cannot be the immediate neighbor, thereby effectively reducing the available pool for subsequent choices.

To simplify the selection process, let's consider the 20 students and the necessity of avoiding consecutive selections. If we denote the three students selected as (x_1, x_2,) and (x_3), we can introduce variables to account for the gaps between selections. Let:

• (y_1 = x_1) (the position of the first selected student),

• (y_2 = x_2 - 1) (the position of the second selected student minus one to account for the gap),

• (y_3 = x_3 - 2) (the position of the third selected student minus two to account for the gaps before the second and third selections).

Now, (y_1, y_2,) and (y_3) must satisfy the following conditions:

1. (y_1 <