If the number of 7 digit numbers whose sum of the digits is equal to 10 and which is formed by using the digits 1, 2 and 3 only is K, then the value of `(K+46)/(100)` is

Answer

To solve the problem of finding the number of 7-digit numbers formed using the digits 1, 2, and 3 such that the sum of the digits equals 10, we can follow these steps:

### Step 1: Set Up the Equation
We need to find the number of non-negative integer solutions to the equation:
\[ x_1 + x_2 + x_3 = 7 \]
where \( x_1 \) is the number of 1's, \( x_2 \) is the number of 2's, and \( x_3 \) is the number of 3's, and the total sum of the digits must equal 10:
\[ 1 \cdot x_1 + 2 \cdot x_2 + 3 \cdot x_3 = 10 \]

### Step 2: Express in Terms of One Variable
From the first equation, we can express \( x_3 \) in terms of \( x_1 \) and \( x_2 \):
\[ x_3 = 7 - x_1 - x_2 \]

Substituting this into the second equation gives:
\[ x_1 + 2x_2 + 3(7 - x_1 - x_2) = 10 \]
Simplifying this:
\[ x_1 + 2x_2 + 21 - 3x_1 - 3x_2 = 10 \]
\[ -2x_1 - x_2 + 21 = 10 \]
\[ -2x_1 - x_2 = -11 \]
\[ 2x_1 + x_2 = 11 \]

### Step 3: Find Non-negative Solutions
Now we need to find non-negative integer solutions to the equation:
\[ 2x_1 + x_2 = 11 \]
From this equation, we can express \( x_2 \) in terms of \( x_1 \):
\[ x_2 = 11 - 2x_1 \]

### Step 4: Determine Valid Values for \( x_1 \)
Since \( x_2 \) must be non-negative:
\[ 11 - 2x_1 \geq 0 \]
This implies:
\[ 2x_1 \leq 11 \]
\[ x_1 \leq 5.5 \]
Thus, \( x_1 \) can take values from 0 to 5 (i.e., \( x_1 = 0, 1, 2, 3, 4, 5 \)).

### Step 5: Calculate Corresponding \( x_2 \) and \( x_3 \)
For each valid \( x_1 \), we can find \( x_2 \) and \( x_3 \):
- If \( x_1 = 0 \): \( x_2 = 11 \), \( x_3 = -4 \) (not valid)
- If \( x_1 = 1 \): \( x_2 = 9 \), \( x_3 = -3 \) (not valid)
- If \( x_1 = 2 \): \( x_2 = 7 \), \( x_3 = -2 \) (not valid)
- If \( x_1 = 3 \): \( x_2 = 5 \), \( x_3 = -1 \) (not valid)
- If \( x_1 = 4 \): \( x_2 = 3 \), \( x_3 = 0 \) (valid)
- If \( x_1 = 5 \): \( x_2 = 1 \), \( x_3 = 0 \) (valid)

### Step 6: Count the Valid Combinations
The valid combinations are:
1. \( (x_1, x_2, x_3) = (4, 3, 0) \)
2. \( (x_1, x_2, x_3) = (5, 1, 0) \)

Now we calculate the number of arrangements for each case:
- For \( (4, 3, 0) \): The number of arrangements is given by:
\[ \frac{7!}{4!3!} = \frac{5040}{24 \times 6} = 35 \]
- For \( (5, 1, 0) \): The number of arrangements is given by:
\[ \frac{7!}{5!1!1!} = \frac{5040}{120 \times 1} = 42 \]

### Step 7: Total Combinations
Thus, the total number \( K \) is:
\[ K = 35 + 42 = 77 \]

### Step 8: Calculate the Final Value
Finally, we need to find the value of \( \frac{K + 46}{100} \):
\[ \frac{77 + 46}{100} = \frac{123}{100} = 1.23 \]

### Final Answer
The value of \( \frac{K + 46}{100} \) is \( 1.23 \).

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