Questions: A certain system has two components. There are 7 different models of the first component and 9 different models of the second. Any first component can be paired with any second component. A salesman must select 2 of the first component and 3 of the second to take on a sales call. How many different sets of components can the salesman take? A) 7 C2 * 9 C3 B) 7 P2 * 9 P3 C) 7 P5 * 9 P5 D) 7 C5 * 9 C5

Solution Steps

To solve this problem, we need to calculate the number of ways to choose 2 models from the 7 available models of the first component and 3 models from the 9 available models of the second component. This is a combination problem, where we use the combination formula \( nCr = \frac{n!}{r!(n-r)!} \) to find the number of ways to choose r items from n items without regard to order.

1. Calculate the number of ways to choose 2 models from 7 models of the first component using combinations.
2. Calculate the number of ways to choose 3 models from 9 models of the second component using combinations.
3. Multiply the two results to get the total number of different sets of components.

To find the number of ways to choose 2 models from 7 models of the first component, we use the combination formula:

\[ {7 \choose 2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

To find the number of ways to choose 3 models from 9 models of the second component, we use the combination formula:

\[ {9 \choose 3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

The total number of different sets of components is the product of the combinations from the first and second components:

\[ 21 \times 84 = 1764 \]

Final Answer