There are three rooms in a hotel: one single, one double and one for four persons. How many ways are there to house seven persons in these rooms?

Options

A. 7!/1!2!4
B. 105
C. 7!/3
D. 7!/3!

Answer

To solve the problem of housing 7 persons in three different types of rooms (1 single room, 1 double room, and 1 room for 4 persons), we can follow these steps:

### Step 1: Choose a person for the single room
We need to select 1 person out of the 7 to occupy the single room. The number of ways to choose 1 person from 7 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of persons and \( r \) is the number of persons to choose.

\[ \text{Ways to choose 1 person for the single room} = \binom{7}{1} = 7 \]

### Step 2: Choose persons for the double room
After selecting 1 person for the single room, we have 6 persons left. Now, we need to choose 2 persons from these 6 to occupy the double room.

\[ \text{Ways to choose 2 persons for the double room} = \binom{6}{2} = 15 \]

### Step 3: Assign the remaining persons to the room for 4 persons
After selecting persons for the single and double rooms, we have 4 persons remaining. All of these 4 persons will occupy the room for 4 persons. The number of ways to choose all 4 persons from 4 is:

\[ \text{Ways to choose 4 persons for the room for 4} = \binom{4}{4} = 1 \]

### Step 4: Calculate the total number of ways
Now, we can find the total number of ways to house the 7 persons by multiplying the number of ways from each step:

\[ \text{Total ways} = \binom{7}{1} \times \binom{6}{2} \times \binom{4}{4} = 7 \times 15 \times 1 = 105 \]

Thus, the total number of ways to house 7 persons in the three types of rooms is **105**.