There are unlimited number of identical balls of three different colours. How many arrangements of atmost 7 balls in a row can be made by using them?
(a) 2187 (b) 343 (c) 399 (d) 3279

Options

A. 2187
B. 343
C. 399
D. 3279

Answer

To find the total number of arrangements of at most 7 balls in a row using an unlimited number of identical balls of three different colors, we can follow these steps:

### Step 1: Understand the Problem
We need to find the total arrangements for 1 to 7 balls in a row, where each ball can be one of three colors.

### Step 2: Calculate Arrangements for Each Case
- For **1 ball**: There are 3 choices (one for each color).
\[ \text{Arrangements for 1 ball} = 3^1 = 3 \]

- For **2 balls**: Each ball can independently be one of the three colors.
\[ \text{Arrangements for 2 balls} = 3^2 = 9 \]

- For **3 balls**: Similarly, each ball has 3 choices.
\[ \text{Arrangements for 3 balls} = 3^3 = 27 \]

- For **4 balls**:
\[ \text{Arrangements for 4 balls} = 3^4 = 81 \]

- For **5 balls**:
\[ \text{Arrangements for 5 balls} = 3^5 = 243 \]

- For **6 balls**:
\[ \text{Arrangements for 6 balls} = 3^6 = 729 \]

- For **7 balls**:
\[ \text{Arrangements for 7 balls} = 3^7 = 2187 \]

### Step 3: Sum the Arrangements
Now, we need to sum all the arrangements from 1 to 7 balls:
\[ \text{Total arrangements} = 3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7 \]
Calculating this gives:
\[ = 3 + 9 + 27 + 81 + 243 + 729 + 2187 \]

### Step 4: Calculate the Total
Now, let's add these values:
\[ 3 + 9 = 12 \]
\[ 12 + 27 = 39 \]
\[ 39 + 81 = 120 \]
\[ 120 + 243 = 363 \]
\[ 363 + 729 = 1092 \]
\[ 1092 + 2187 = 3279 \]

### Final Answer
Thus, the total number of arrangements of at most 7 balls in a row is:
\[ \boxed{3279} \]