Key Concepts and Formulas

• Stars and Bars Theorem: The number of non-negative integer solutions to the equation $x_1 + x_2 + ... + x_k = n$ is given by ${n+k-1 \choose k-1} = {n+k-1 \choose n}$.
• Variable Substitution: When variables have lower bound restrictions (e.g., $x \ge a$), we can substitute new variables to transform the problem into one with non-negative integer solutions.

Step-by-Step Solution

Step 1: State the given equation and constraints

We are given the equation $x+y+z=21$, with the constraints $x \ge 1$, $y \ge 3$, and $z \ge 4$.

Step 2: Transform the variables to satisfy non-negativity

To apply the Stars and Bars theorem, we need non-negative integer solutions. We introduce new variables $x_1$, $y_1$, and $z_1$ such that:

• $x = x_1 + 1$, where $x_1 \ge 0$
• $y = y_1 + 3$, where $y_1 \ge 0$
• $z = z_1 + 4$, where $z_1 \ge 0$

This ensures that $x \ge 1$, $y \ge 3$, and $z \ge 4$.

Step 3: Substitute the new variables into the original equation

Substitute the expressions for $x$, $y$, and $z$ in terms of $x_1$, $y_1$, and $z_1$ into the original equation: $(x_1 + 1) + (y_1 + 3) + (z_1 + 4) = 21$

Step 4: Simplify the equation

Simplify the equation by combining the constants: $x_1 + y_1 + z_1 + 1 + 3 + 4 = 21$ $x_1 + y_1 + z_1 + 8 = 21$ $x_1 + y_1 + z_1 = 21 - 8$ $x_1 + y_1 + z_1 = 13$

Now we have an equation with non-negative integer solutions, where $x_1 \ge 0$, $y_1 \ge 0$, and $z_1 \ge 0$.

Step 5: Apply the Stars and Bars Theorem

We can now apply the Stars and Bars theorem to find the number of non-negative integer solutions to $x_1 + y_1 + z_1 = 13$. Here, $n = 13$ and $k = 3$. The number of solutions is given by: ${n+k-1 \choose k-1} = {13+3-1 \choose 3-1} = {15 \choose 2}$

Step 6: Calculate the binomial coefficient

Calculate the binomial coefficient: ${15 \choose 2} = \frac{15!}{2!(15-2)!} = \frac{15!}{2!13!} = \frac{15 \times 14}{2 \times 1} = 15 \times 7 = 105$

Common Mistakes & Tips

• Forgetting to adjust the variables: The most common mistake is forgetting to introduce new variables to account for the lower bound restrictions on the original variables. Always transform the variables to ensure they are non-negative.
• Incorrectly applying the formula: Make sure you correctly identify $n$ and $k$ in the Stars and Bars formula. $n$ is the sum the variables must add up to, and $k$ is the number of variables.
• Double-checking constraints: Always double-check that the new variables satisfy the non-negativity constraints after the substitution.

Summary

We are given the equation $x+y+z=21$ with the constraints $x \ge 1, y \ge 3, z \ge 4$. By substituting $x = x_1 + 1$, $y = y_1 + 3$, and $z = z_1 + 4$, we transform the equation into $x_1 + y_1 + z_1 = 13$ with non-negative integer solutions. Applying the Stars and Bars theorem, the number of solutions is ${13+3-1 \choose 3-1} = {15 \choose 2} = 105$.

Final Answer

The final answer is $\boxed{105}$.