In $a$ group of 8 people, how many ways can you choose $a$ committee of 3 people if two specific individuals refuse to work together?

Step 1 (Total Unrestricted Combinations): We want to choose a committee of 3 people from a group of 8. Without any restrictions, the number of ways is:

Step 2 (Identify Invalid Combinations): Let the two individuals who refuse to work together be $X$ and $Y$. A committee is invalid if it contains both $X$ and $Y$.

• Since $X$ and $Y$ occupy 2 seats in the 3-person committee, we must choose exactly 1 more person from the remaining $8-2=6$ people to fill the last seat.

• The number of invalid committees is:

Step 3 (Calculate Valid Combinations): We subtract the invalid committees from the total unrestricted committees:

Step 4 (Conclusion): The committee can be chosen in exactly 50 different ways.