H2O is chain of salons, employing one hair stylist at each salon. It has got centralized system in place to provide appointments through phone calls for all its 6 salons located in different areas in Pune. A family of four seeking the appointment for all its members for the hair cut. On a particular day, in how many ways the salon can give the appointments to them?

Options

A. A) 3024
B. B) 2524
C. C) 3384
D. D) 3084

Answer

To solve the problem of how many ways the salon can give appointments to a family of four members seeking haircuts at six different salons, we can follow these steps:

### Step 1: Understand the Problem
We have a family of four members (let’s call them A, B, C, and D) who want to get haircuts at any of the six salons. Each salon can accommodate one appointment at a time.

### Step 2: Define the Variables
Let’s denote the number of appointments given to each salon as follows:
- \( x_1 \): Number of appointments at Salon 1
- \( x_2 \): Number of appointments at Salon 2
- \( x_3 \): Number of appointments at Salon 3
- \( x_4 \): Number of appointments at Salon 4
- \( x_5 \): Number of appointments at Salon 5
- \( x_6 \): Number of appointments at Salon 6

We need to find the non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 4 \]
where \( x_i \) represents the number of family members getting appointments at salon \( i \).

### Step 3: Apply the Stars and Bars Theorem
The problem can be solved using the "stars and bars" theorem, which states that the number of non-negative integer solutions to the equation \( x_1 + x_2 + ... + x_r = n \) is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number of items to distribute (in this case, haircuts) and \( r \) is the number of categories (in this case, salons).

In our case:
- \( n = 4 \) (the four family members)
- \( r = 6 \) (the six salons)

### Step 4: Calculate the Combinations
Using the formula:
\[ \binom{n + r - 1}{r - 1} = \binom{4 + 6 - 1}{6 - 1} = \binom{9}{5} \]

Now, calculate \( \binom{9}{5} \):
\[ \binom{9}{5} = \frac{9!}{5!(9-5)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = \frac{3024}{24} = 126 \]

### Step 5: Account for the Arrangement of Family Members
Since the four family members can be arranged among themselves in \( 4! \) (factorial of 4) ways, we need to multiply the number of combinations by the number of arrangements:
\[ 4! = 24 \]

Thus, the total number of ways to give appointments is:
\[ 126 \times 24 = 3024 \]

### Final Answer
The total number of ways the salon can give appointments to the family of four is **3024**.

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