How do you account for a variable's lower bound when finding nonnegative integer solutions?

Conditions

• One or more variables have a lower bound greater than zero.
• The other variables are nonnegative.
• The equation is of the form $x_1+\cdots+x_n=r$.

Reasoning, step by step

1. Identify the variable with a lower bound (e.g., $x_3 \ge 2$).
2. Substitute a new variable to shift the lower bound to zero (e.g., $x_3' = x_3 - 2$).
3. Adjust the total sum $r$ by subtracting the lower bounds.
4. Solve the new equation for nonnegative integer solutions using stars and bars.

Example

For $x_1+x_2+x_3=9$ with $x_3\ge2$ and the others nonnegative, substitute $x_3'=x_3-2$. The remaining total is 7, giving $\binom{9}{2}=36$ solutions.

Common misconceptions

• Forgetting to adjust the total sum $r$ after shifting the variable.
• Applying the standard stars and bars formula without accounting for the lower bound.

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