Key Concepts and Formulas

• Linear Diophantine Equations: These are linear equations where the solutions are required to be integers. In this problem, we need to find non-negative integer solutions.
• Permutations with Repetition: If we have $n$ objects where $n_1$ are of one type, $n_2$ are of a second type, ..., and $n_k$ are of a $k$-th type, then the number of distinct arrangements (permutations) of these $n$ objects is given by: $\frac{n!}{n_1! n_2! \cdots n_k!}$ where $n = n_1 + n_2 + \cdots + n_k$.
• Factorial: For a non-negative integer $n$, the factorial $n!$ is the product of all positive integers less than or equal to $n$. By convention, $0! = 1$.

Step-by-Step Solution

Step 1: Define Variables and Set Up Equations

We define the variables to represent the number of times each digit appears in the seven-digit number. Let:

• $x$ be the number of times the digit '1' appears.
• $y$ be the number of times the digit '2' appears.
• $z$ be the number of times the digit '3' appears.

Since we are forming a seven-digit number, the sum of the counts of each digit must be 7: $x + y + z = 7 \quad \text{(Equation 1)}$

The sum of the digits must be 11, so we have: $1x + 2y + 3z = 11 \quad \text{(Equation 2)}$

We seek non-negative integer solutions for $x, y,$ and $z$.

Step 2: Simplify the Equations

Subtract Equation 1 from Equation 2 to eliminate $x$: $(x + 2y + 3z) - (x + y + z) = 11 - 7$ $y + 2z = 4 \quad \text{(Equation 3)}$

Step 3: Find Possible Values for z

From Equation 3, we can express $y$ in terms of $z$: $y = 4 - 2z$. Since $y$ must be non-negative, we have $4 - 2z \geq 0$, which implies $2z \leq 4$, or $z \leq 2$. Therefore, the possible non-negative integer values for $z$ are 0, 1, and 2.

Step 4: Case Analysis for Each Value of z

We will analyze each possible value of $z$ and find the corresponding values of $x$ and $y$.

• Case 1: $z = 0$

  • Substituting $z = 0$ into Equation 3, we get $y = 4 - 2(0) = 4$.
  • Substituting $y = 4$ and $z = 0$ into Equation 1, we get $x + 4 + 0 = 7$, so $x = 3$.
  • Thus, we have $x = 3, y = 4, z = 0$. The digits are {1, 1, 1, 2, 2, 2, 2}.
  • The number of distinct arrangements is $\frac{7!}{3!4!0!} = \frac{7 \cdot 6 \cdot 5}{3 \cdot 2 \cdot 1} = 35$.

• Case 2: $z = 1$

  • Substituting $z = 1$ into Equation 3, we get $y = 4 - 2(1) = 2$.
  • Substituting $y = 2$ and $z = 1$ into Equation 1, we get $x + 2 + 1 = 7$, so $x = 4$.
  • Thus, we have $x = 4, y = 2, z = 1$. The digits are {1, 1, 1, 1, 2, 2, 3}.
  • The number of distinct arrangements is $\frac{7!}{4!2!1!} = \frac{7 \cdot 6 \cdot 5}{2 \cdot 1} = 105$.

• Case 3: $z = 2$

  • Substituting $z = 2$ into Equation 3, we get $y = 4 - 2(2) = 0$.
  • Substituting $y = 0$ and $z = 2$ into Equation 1, we get $x + 0 + 2 = 7$, so $x = 5$.
  • Thus, we have $x = 5, y = 0, z = 2$. The digits are {1, 1, 1, 1, 1, 3, 3}.
  • The number of distinct arrangements is $\frac{7!}{5!0!2!} = \frac{7 \cdot 6}{2 \cdot 1} = 21$.

Step 5: Calculate the Total Number of Elements

The total number of elements in set $P$ is the sum of the distinct arrangements from each case: $\text{Total numbers} = 35 + 105 + 21 = 161$

Common Mistakes & Tips

• Ensure that you consider all possible values for the variable you are using to iterate through the cases. Remember that the values must be non-negative integers.
• Double-check your factorial calculations to avoid errors in computing the number of distinct arrangements for each case.
• Always verify that the sum of the counts of each digit equals 7 and that the sum of the digits equals 11 for each case.

Summary

We found the number of seven-digit numbers that can be formed using the digits 1, 2, and 3 such that the sum of their digits is 11 by setting up and solving a system of linear Diophantine equations. We then used the formula for permutations with repetition to calculate the number of distinct arrangements for each possible combination of digits. Summing the number of arrangements for each case gave us the total number of such numbers, which is 161.

The final answer is $\boxed{161}$, which corresponds to option (C).