Let \( P \) be the set of seven-digit numbers with the sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2, and 3 only, then the number of elements in the set \( P \) is:
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To solve problems of this nature, carefully set up a Diophantine equation based on the constraints, and use appropriate combinatorial methods to find the number of solutions.
Step 1: Determine the count of seven-digit numbers formed from digits 1, 2, and 3, such that the sum of these digits equals 11.
Step 2: Formulate the problem as solving the system of equations: \( x_1 + x_2 + x_3 = 7 \) and \( 1x_1 + 2x_2 + 3x_3 = 11 \), where \( x_1, x_2, x_3 \) denote the frequencies of digits 1, 2, and 3, respectively.
Step 3: Employ combinatorial methods or generating functions to find the number of non-negative integer solutions for the Diophantine equation system.
Step 4: The total number of valid seven-digit numbers is 161. The correct option is (3).
