Step-by-step Solution:

Step 1: Problem Setup We need to find the number of solutions to the equation: \[ x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 = 10 \] where each \( x_i \) (digit in the 7-digit number) satisfies \( 1 \leq x_i \leq 3 \). --- Step 2: Transform the Problem To simplify, let \( y_i = x_i - 1 \). Then \( y_i \geq 0 \), and the equation becomes: \[ (y_1 + 1) + (y_2 + 1) + \cdots + (y_7 + 1) = 10 \] \[ y_1 + y_2 + \cdots + y_7 = 3 \] Now, we need to find the number of non-negative integer solutions to \( y_1 + y_2 + \cdots + y_7 = 3 \), with \( y_i \leq 2 \) (since \( x_i \leq 3 \)). --- Step 3: Use Stars and Bars The total number of non-negative integer solutions to \( y_1 + y_2 + \cdots + y_7 = 3 \) is: \[ \binom{3 + 7 - 1}{7 - 1} = \binom{9}{6} = 84 \] --- Step 4: Subtract Invalid Solutions We must subtract the cases where \( y_i \geq 3 \). Suppose \( y_1 \geq 3 \). Let \( y_1' = y_1 - 3 \). Then the equation becomes: \[ y_1' + y_2 + \cdots + y_7 = 0 \] The number of solutions is: \[ \binom{0 + 7 - 1}{7 - 1} = \binom{6}{6} = 1 \] Since any of the 7 variables could be \( \geq 3 \), there are \( 7 \times 1 = 7 \) invalid solutions. --- Step 5: Final Calculation Subtract the invalid solutions from the total: \[ 84 - 7 = 77 \] --- Step 6: Conclusion The number of 7-digit numbers formed using the digits 1, 2, and 3 only, with the sum of the digits equal to 10, is: \[ \boxed{77} \] Correct Answer: \(\boxed{C}\)