Orange Computers is breaking up its conference attendees into groups. Each group must have exactly 1 person from Division A, 2 people from Division B, and 3 people from Division C. There are 20 people from Division A, 30 people from Division B, and 40 people from Division C at the conference. What is the smallest number of people who will not be able to be assigned to a group?

Options

A. 12
B. 5
C. 2
D. 1

Answer

To solve the problem of how many people will not be able to be assigned to a group, we need to follow these steps:

### Step 1: Understand the Group Composition
Each group must consist of:
- 1 person from Division A
- 2 people from Division B
- 3 people from Division C

### Step 2: Determine the Total Number of People in Each Division
- Division A: 20 people
- Division B: 30 people
- Division C: 40 people

### Step 3: Calculate the Maximum Number of Groups from Each Division
- **From Division A:** Since each group requires 1 person, the maximum number of groups that can be formed is:
\[ \text{Groups from A} = 20 \text{ people} \div 1 = 20 \text{ groups} \]

- **From Division B:** Since each group requires 2 people, the maximum number of groups that can be formed is:
\[ \text{Groups from B} = 30 \text{ people} \div 2 = 15 \text{ groups} \]

- **From Division C:** Since each group requires 3 people, the maximum number of groups that can be formed is:
\[ \text{Groups from C} = 40 \text{ people} \div 3 = 13 \text{ groups} \quad (\text{since } 40 \div 3 = 13 \text{ remainder } 1) \]

### Step 4: Identify the Limiting Factor
The limiting factor for the number of groups is Division C, which can only form 13 groups. Thus, we can only form 13 groups in total.

### Step 5: Calculate the Number of People Assigned to Each Division
- **From Division A:**
\[ 13 \text{ groups} \times 1 \text{ person/group} = 13 \text{ people assigned} \]
Remaining in Division A:
\[ 20 - 13 = 7 \text{ people unassigned} \]

- **From Division B:**
\[ 13 \text{ groups} \times 2 \text{ people/group} = 26 \text{ people assigned} \]
Remaining in Division B:
\[ 30 - 26 = 4 \text{ people unassigned} \]

- **From Division C:**
\[ 13 \text{ groups} \times 3 \text{ people/group} = 39 \text{ people assigned} \]
Remaining in Division C:
\[ 40 - 39 = 1 \text{ person unassigned} \]

### Step 6: Calculate the Total Number of Unassigned People
Now, we sum the unassigned people from all divisions:
\[ \text{Total unassigned} = 7 \text{ (from A)} + 4 \text{ (from B)} + 1 \text{ (from C)} = 12 \]

### Final Answer
The smallest number of people who will not be able to be assigned to a group is **12**.

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