### Related Formula
For a linear equation
x + 2y = k$x + 2y = k$, where
x, y ge 0$x, y \ge 0$ are non-negative integers, the number of distinct integral pairs
(x,y)$(x,y)$ is equal to the number of possible non-negative values that
y$y$ can take, which is given by:
leftlfloor frack2 rightrfloor + 1$$\left\lfloor \frac{k}{2} \right\rfloor + 1$$
### Core Logic
We need to find the number of
non-negative integer solutions to:
x + 2y + 3z = 42$$x + 2y + 3z = 42$$
Rearranging the equation to analyze values branch-by-branch based on
z$z$:
x + 2y = 42 - 3z$$x + 2y = 42 - 3z$$
Since
x, y ge 0$x, y \ge 0$, the maximum value
z$z$ can attain corresponds to
x=0, y=0$x=0, y=0$, so
3z le 42 implies z le 14$3z \le 42 \implies z \le 14$.
### Step 1: Enumerate Values across all z configurations
Let's count the possible solutions for each integer value of
z$z$ from
0$0$ to
14$14$:
-
z = 0 implies x + 2y = 42 implies lfloor 42/2 rfloor + 1 = 22 text solutions$z = 0 \implies x + 2y = 42 \implies \lfloor 42/2 \rfloor + 1 = 22 \text{ solutions}$
-
z = 1 implies x + 2y = 39 implies lfloor 39/2 rfloor + 1 = 20 text solutions$z = 1 \implies x + 2y = 39 \implies \lfloor 39/2 \rfloor + 1 = 20 \text{ solutions}$
-
z = 2 implies x + 2y = 36 implies lfloor 36/2 rfloor + 1 = 19 text solutions$z = 2 \implies x + 2y = 36 \implies \lfloor 36/2 \rfloor + 1 = 19 \text{ solutions}$
-
z = 3 implies x + 2y = 33 implies lfloor 33/2 rfloor + 1 = 17 text solutions$z = 3 \implies x + 2y = 33 \implies \lfloor 33/2 \rfloor + 1 = 17 \text{ solutions}$
-
z = 4 implies x + 2y = 30 implies lfloor 30/2 rfloor + 1 = 16 text solutions$z = 4 \implies x + 2y = 30 \implies \lfloor 30/2 \rfloor + 1 = 16 \text{ solutions}$
-
z = 5 implies x + 2y = 27 implies lfloor 27/2 rfloor + 1 = 14 text solutions$z = 5 \implies x + 2y = 27 \implies \lfloor 27/2 \rfloor + 1 = 14 \text{ solutions}$
-
z = 6 implies x + 2y = 24 implies lfloor 24/2 rfloor + 1 = 13 text solutions$z = 6 \implies x + 2y = 24 \implies \lfloor 24/2 \rfloor + 1 = 13 \text{ solutions}$
-
z = 7 implies x + 2y = 21 implies lfloor 21/2 rfloor + 1 = 11 text solutions$z = 7 \implies x + 2y = 21 \implies \lfloor 21/2 \rfloor + 1 = 11 \text{ solutions}$
-
z = 8 implies x + 2y = 18 implies lfloor 18/2 rfloor + 1 = 10 text solutions$z = 8 \implies x + 2y = 18 \implies \lfloor 18/2 \rfloor + 1 = 10 \text{ solutions}$
-
z = 9 implies x + 2y = 15 implies lfloor 15/2 rfloor + 1 = 8 text solutions$z = 9 \implies x + 2y = 15 \implies \lfloor 15/2 \rfloor + 1 = 8 \text{ solutions}$
-
z = 10 implies x + 2y = 12 implies lfloor 12/2 rfloor + 1 = 7 text solutions$z = 10 \implies x + 2y = 12 \implies \lfloor 12/2 \rfloor + 1 = 7 \text{ solutions}$
-
z = 11 implies x + 2y = 9 implies lfloor 9/2 rfloor + 1 = 5 text solutions$z = 11 \implies x + 2y = 9 \implies \lfloor 9/2 \rfloor + 1 = 5 \text{ solutions}$
-
z = 12 implies x + 2y = 6 implies lfloor 6/2 rfloor + 1 = 4 text solutions$z = 12 \implies x + 2y = 6 \implies \lfloor 6/2 \rfloor + 1 = 4 \text{ solutions}$
-
z = 13 implies x + 2y = 3 implies lfloor 3/2 rfloor + 1 = 2 text solutions$z = 13 \implies x + 2y = 3 \implies \lfloor 3/2 \rfloor + 1 = 2 \text{ solutions}$
-
z = 14 implies x + 2y = 0 implies lfloor 0/2 rfloor + 1 = 1 text solution$z = 14 \implies x + 2y = 0 \implies \lfloor 0/2 \rfloor + 1 = 1 \text{ solution}$
### Step 2: Total Sum Computation
Summing all the calculated distribution counts:
textTotal solutions = 22 + 20 + 19 + 17 + 16 + 14 + 13 + 11 + 10 + 8 + 7 + 5 + 4 + 2 + 1 = 169$$\text{Total solutions} = 22 + 20 + 19 + 17 + 16 + 14 + 13 + 11 + 10 + 8 + 7 + 5 + 4 + 2 + 1 = 169$$
### Pattern Recognition
Sees: Linear multi-variable diophantine solution constraints.
Shortcut: Grouping into alternating arithmetic sequences can expedite the total calculation step instead of adding each discrete integer row manually.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Mathematics: Permutations and Combinations
