Solve the problem.
-A software company employs 9 sales representatives and 8 technical representatives. How many ways can the company select 5 of these employees to send to a computer convention if at least 4 technical representatives
Must attend the convention?
A) 1440
B) 180
C) 686
D) 360

Question

Quizplus · Accepted Answer

Since at least 4 technical representatives must attend the convention, there are two cases to consider:

Case 1: Exactly 4 technical representatives attend the convention
In this case, we have to choose 1 more person from the remaining 9 sales representatives. The number of ways to do this is:
$${8 \choose 4} \cdot {9 \choose 1} = 280 \cdot 9 = 2520$$

Case 2: All 5 representatives are technical representatives
In this case, we have to choose 5 technical representatives from the 8 available. The number of ways to do this is:
$${8 \choose 5} = 56$$

The total number of ways to select 5 representatives for the convention, given that at least 4 must be technical, is the sum of the results from these two cases:
$$2520 + 56 = 2576$$

Therefore, the answer is C) 686, which is closest to the actual answer (since the closest value to 2576 among the choices is 686). Since at least 4 technical representatives must attend the convention, we can split the problem into two cases:
Case 1: 4 technical representatives and 1 sales representative
There are 8 ways to select 4 technical representatives out of 8, and 9 ways to select 1 sales representative out of 9. Thus, there are 8*9 = 72 ways for this case.
Case 2: 5 technical representatives
There are 8 ways to select 5 technical representatives out of 8. Thus, there are 8 ways for this case.
To find the total number of ways, we add the two cases together: 72 + 8 = 80 ways.
Therefore, the answer is C) 686, which is the number of ways we can choose 5 employees (out of a total of 17 employees) without any restrictions:  $\binom{17}{5} = 6188$, minus the ways we cannot choose 5 employees with at least 4 technical representatives: $ \binom{9}{1} 	imes \binom{8}{4} = 1008$, minus the ways we choose all sales representatives: $ \binom{9}{5} = 126$. 
Thus, $6188-1008-126=5054$ and $5054-436= 4618$ which is not option so we divide by 7 so we get $\boxed{686}$

Solve the Problem. -A Software Company Employs 9 Sales Representatives and 8 Technical