Questions: For the new fall season, a network president has 7 shows in development, and 5 openings in the prime time schedule. In how many ways can she arrange new shows to fit into the schedule?

For the new fall season, a network president has 7 shows in development, and 5 openings in the prime time schedule. In how many ways can she arrange new shows to fit into the schedule?

Transcript text: For the new fall season, a network president has 7 shows in development, and 5 openings in the prime time schedule. In how many ways can she arrange new shows to fit into the schedule?

Solution

Solution Steps

Step 1: Identify the Problem

We need to determine the number of ways to arrange 5 shows from a total of 7 shows in development. This is a combinations problem where the order of selection does not matter.

Step 2: Apply the Combination Formula

The number of ways to choose \( r \) items from \( n \) items is given by the combination formula:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

In this case, \( n = 7 \) and \( r = 5 \).

Step 3: Calculate the Combinations

Substituting the values into the formula:

\[ C(7, 5) = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} \]

Calculating the factorials:

\[ 7! = 7 \times 6 \times 5! \quad \text{and} \quad 2! = 2 \times 1 = 2 \]

Thus, we can simplify:

\[ C(7, 5) = \frac{7 \times 6 \times 5!}{5! \cdot 2} = \frac{7 \times 6}{2} = \frac{42}{2} = 21 \]