Five people are distributed in four identical rooms. A room can also contain zero people. Find the number of ways to distribute then

Options

A. 47
B. 53
C. 43
D. 51

Answer

To solve the problem of distributing 5 people into 4 identical rooms where a room can also contain zero people, we can use the "stars and bars" combinatorial method along with some case analysis. Here's a step-by-step breakdown of the solution:

### Step 1: Understanding the Problem
We need to distribute 5 indistinguishable people into 4 identical rooms. Since the rooms are identical, the order in which we place the people in the rooms does not matter.

### Step 2: Using the Stars and Bars Method
We can represent the distribution of people as a problem of placing "stars" (people) into "bars" (rooms). However, since the rooms are identical, we will consider different distributions based on the number of people in each room.

### Step 3: Case Analysis
We will analyze the different cases based on how many people are in each room.

#### Case 1: All 5 people in one room
- Distribution: (5, 0, 0, 0)
- Number of ways: 1 way

#### Case 2: 4 people in one room, 1 in another
- Distribution: (4, 1, 0, 0)
- Number of ways: 1 way

#### Case 3: 3 people in one room, 2 in another
- Distribution: (3, 2, 0, 0)
- Number of ways: 1 way

#### Case 4: 3 people in one room, 1 in another, 1 in another
- Distribution: (3, 1, 1, 0)
- Number of ways: 1 way (since the two rooms with 1 person are identical)

#### Case 5: 2 people in one room, 2 in another
- Distribution: (2, 2, 0, 0)
- Number of ways: 1 way

#### Case 6: 2 people in one room, 1 in another, 1 in another, 1 in another
- Distribution: (2, 1, 1, 1)
- Number of ways: 1 way

### Step 4: Summing Up the Cases
Now we will sum the number of ways from all the cases:
- Case 1: 1 way
- Case 2: 1 way
- Case 3: 1 way
- Case 4: 1 way
- Case 5: 1 way
- Case 6: 1 way

Total number of ways = 1 + 1 + 1 + 1 + 1 + 1 = 6 ways

### Step 5: Final Answer
Thus, the total number of ways to distribute 5 people into 4 identical rooms is **6 ways**.