Question

The number of non-negative integral solutions of \( x_1 + x_2 + x_3 + x_4 = 10 \) is:

• A. 120
• B. 144
• C. 256
• D. 286

Hint:

For "stars and bars" problems, use the formula \( \binom{n + k - 1}{k - 1} \) to calculate the number of solutions.

Answer 1

The Correct Option is D

We are asked to find the number of non-negative integral solutions to the equation: \[ x_1 + x_2 + x_3 + x_4 = 10 \] This is a classic example of a "stars and bars" problem, where we need to distribute 10 indistinguishable stars (the total sum) into 4 distinguishable bins (the variables \( x_1, x_2, x_3, x_4 \)). The formula for the number of solutions to the equation \( x_1 + x_2 + \dots + x_k = n \) where \( x_1, x_2, \dots, x_k \) are non-negative integers is: \[ \binom{n + k - 1}{k - 1} \] In our case, \( n = 10 \) and \( k = 4 \), so the number of solutions is: \[ \binom{10 + 4 - 1}{4 - 1} = \binom{13}{3} \] Now, calculate \( \binom{13}{3} \): \[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \] Thus, the number of non-negative integral solutions is 286, which corresponds to option (4).