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 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 \( \boxed{15} \).