To find the number of combinations of shows that Gabe can choose to watch, we need to calculate the number of ways to choose 4 shows from a total of 6 shows.
This can be done using the combination formula: \[ C(n, r) = \frac{n!}{r!(n - r)!} \] where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial.
In this case, \( n = 6 \) (the total number of shows) and \( r = 4 \) (the number of shows he can watch).
Now, we calculate \( C(6, 4) \): \[ C(6, 4) = \frac{6!}{4!(6 - 4)!} = \frac{6!}{4! \cdot 2!} \]
Calculate the factorials:
- \( 6! = 720 \)
- \( 4! = 24 \)
- \( 2! = 2 \)
Now we can substitute these values into the formula: \[ C(6, 4) = \frac{720}{24 \cdot 2} = \frac{720}{48} = 15 \]
Thus, the number of different combinations of shows that Gabe can choose to watch is \( \boxed{15} \).