The number of eight - digit integers, with the sum of digits equal to 12 and formed by using of digits 1, 2 and 3 only are

Options

A. 255
B. 277
C. 288
D. 266

Answer

To find the number of eight-digit integers formed using the digits 1, 2, and 3, with the sum of the digits equal to 12, we can approach the problem systematically.

### Step 1: Define the Variables
Let:
- \( x_1 \) = number of 1's used
- \( x_2 \) = number of 2's used
- \( x_3 \) = number of 3's used

We need to satisfy the following conditions:
1. \( x_1 + x_2 + x_3 = 8 \) (total digits)
2. \( x_1 + 2x_2 + 3x_3 = 12 \) (sum of digits)

### Step 2: Solve the Equations
From the first equation, we can express \( x_1 \) in terms of \( x_2 \) and \( x_3 \):
\[ x_1 = 8 - x_2 - x_3 \]

Substituting \( x_1 \) into the second equation:
\[ (8 - x_2 - x_3) + 2x_2 + 3x_3 = 12 \]
This simplifies to:
\[ 8 + x_2 + 2x_3 = 12 \]
\[ x_2 + 2x_3 = 4 \]

### Step 3: Find Non-negative Integer Solutions
Now we need to find non-negative integer solutions for the equation \( x_2 + 2x_3 = 4 \).

- If \( x_3 = 0 \): \( x_2 = 4 \) → \( (x_1, x_2, x_3) = (4, 4, 0) \)
- If \( x_3 = 1 \): \( x_2 = 2 \) → \( (x_1, x_2, x_3) = (5, 2, 1) \)
- If \( x_3 = 2 \): \( x_2 = 0 \) → \( (x_1, x_2, x_3) = (6, 0, 2) \)

Thus, the valid combinations of \( (x_1, x_2, x_3) \) are:
1. \( (4, 4, 0) \)
2. \( (5, 2, 1) \)
3. \( (6, 0, 2) \)

### Step 4: Calculate the Permutations for Each Case
We will calculate the number of arrangements for each case using the formula for permutations of multiset:
\[ \text{Number of arrangements} = \frac{n!}{x_1! \cdot x_2! \cdot x_3!} \]

1. **Case 1: \( (4, 4, 0) \)**
\[ \text{Arrangements} = \frac{8!}{4! \cdot 4!} = \frac{40320}{24 \cdot 24} = 70 \]

2. **Case 2: \( (5, 2, 1) \)**
\[ \text{Arrangements} = \frac{8!}{5! \cdot 2! \cdot 1!} = \frac{40320}{120 \cdot 2 \cdot 1} = 168 \]

3. **Case 3: \( (6, 0, 2) \)**
\[ \text{Arrangements} = \frac{8!}{6! \cdot 0! \cdot 2!} = \frac{40320}{720 \cdot 1 \cdot 2} = 28 \]

### Step 5: Total Arrangements
Now, we sum the arrangements from all cases:
\[ \text{Total} = 70 + 168 + 28 = 266 \]

### Conclusion
The total number of eight-digit integers formed using the digits 1, 2, and 3, with the sum of the digits equal to 12, is **266**.