To find out how many different combinations of shows Gabe can choose to watch, we need to calculate the number of ways to choose 4 shows from the 6 available shows. This is a combinations problem and can be solved using the binomial coefficient formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \(n\) is the total number of items (shows), and \(r\) is the number of items to choose.
In this case, \(n = 6\) and \(r = 4\). Therefore, we need to calculate:
\[ \binom{6}{4} = \frac{6!}{4!(6-4)!} = \frac{6!}{4! \cdot 2!} \]
Calculating the factorial values:
- \(6! = 6 \times 5 \times 4! = 720\)
- \(4! = 24\)
- \(2! = 2\)
Now plug these back into the equation:
\[ \binom{6}{4} = \frac{720}{24 \cdot 2} = \frac{720}{48} = 15 \]
Thus, the number of different combinations of shows that Gabe can choose to watch is:
\[ \boxed{15} \]