Question

There are 6 television shows coming on this week that Gabe wants to watch. They all come on at different times, but his mom will only let him watch 4 shows a week. How many different combinations of shows can Gabe choose to watch? (1 point) Responses 15 15 30 30 45 45 60

Answer 1

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:

\[
\boxed{15}
\]