Question

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:

• A. 158
• B. 173
• C. 161
• D. 164

Hint:

Use the stars and bars method to count the number of solutions to equations involving non-negative integers.

Answer 1

The Correct Option is C

We need to find the number of seven-digit numbers where the sum of the digits equals 11, and the digits are restricted to 1, 2, and 3. Let the number of 1's be \( x_1 \), the number of 2's be \( x_2 \), and the number of 3's be \( x_3 \). The equation becomes: \[ x_1 + 2x_2 + 3x_3 = 11, \] with the constraint \( x_1 + x_2 + x_3 = 7 \) (since there are seven digits). Solving this system of equations using the stars and bars method, we find the total number of solutions is 161. 
Thus, the number of elements in the set P is 161.