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:
Show Hint
- 158
- 173
- 161
- 164
The Correct Option is C
Solution and Explanation
Step 1: The problem asks us to find the number of seven-digit numbers that can be formed using the digits 1, 2, and 3, where the sum of the digits is 11.
Step 2: The equation we need to solve is \( x_1 + x_2 + x_3 = 7 \) where \( x_1, x_2, x_3 \) represent the number of times the digits 1, 2, and 3 are used, respectively. The constraint is that \( 1x_1 + 2x_2 + 3x_3 = 11 \).
Step 3: This is a Diophantine equation, and we can find the number of non-negative integer solutions that satisfy both conditions using methods such as generating functions or combinatorics.
Step 4: After solving, the number of valid seven-digit numbers is found to be 161. Thus, the correct answer is (3).
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