Question

How many solutions are there to the equation \(x_1 + x_2 + x_3 + x_4 = 17\), where \(x_1, x_2, x_3, x_4\) are nonnegative integers?

• A. 1140
• B. 1160
• C. 1040
• D. 1200

Hint:

The "stars and bars" theorem helps in counting the number of solutions to equations of the form \(x_1 + x_2 + ... + x_n = k\) where \(x_1, x_2, ..., x_n\) are non-negative integers.

Answer 1

The Correct Option is A

This is a classic example of the stars and bars problem, where the total sum is 17 and we are partitioning it into 4 nonnegative integers. The formula for the number of solutions is: \[ \text{Number of solutions} = \binom{17 + 4 - 1}{4 - 1} = \binom{20}{3} \] Now, calculating: \[ \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \] Hence, the number of solutions is 1140.