Solve the problem.
-A home cooking equipment supply company employs 7 sales representatives and 6 product designers. How many ways can this company select 4 of these employees to send to a product demonstration convention in
Oahu if at least 3 product designers must attend?

Question

Quizplus · Accepted Answer

To solve this problem, we need to use a combination of product designers and sales representatives where at least 3 product designers are included. 
Case 1: 3 product designers are selected along with 1 sales representative:
This can be done in ${6\choose3}\cdot{7\choose1} = 140$ ways. 
Case 2: All 4 selected employees are product designers:
This can be done in ${6\choose4} = 15$ ways. 
Total number of ways to select 4 employees with at least 3 product designers = 140 + 15 = 155.
Therefore, the best choice is C. We can split this problem into two cases:
Case 1: 3 product designers are selected
In this case, we need to select one more employee from the remaining 10. This can be done in 10 ways (since we can choose from any of the 7 sales representatives or the remaining 3 product designers).
Number of ways to select 3 product designers = $\binom{6}{3}$ = 20
Total number of ways in Case 1 = 20 x 10 = 200
Case 2: 4 product designers are selected
In this case, we simply need to select 4 product designers from the 6 available. This can be done in $\binom{6}{4}$ = 15 ways.
Total number of ways in Case 2 = 15
Total number of ways to select 4 employees such that at least 3 are product designers = Case 1 + Case 2 = 200 + 15 = 215
Therefore, the answer is C (155).

Solve the Problem. -A Home Cooking Equipment Supply Company Employs 7 Sales Representatives