Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Let $\vec{a} = \vec{i} + \vec{j} + \vec{k}$ and let $\vec{r}$ be a variable vector such that $\vec{r}\cdot\vec{i}, \vec{r}\cdot\vec{j}$ and $\vec{r}\cdot\vec{k}$ are positive integers. If $\vec{r}\cdot\vec{a} \leq 12$ then the number of values of $\vec{r}$ is:
Step-by-Step Solution
Key Concept: Transform the constraint $x + y + z \leq 12$ with positive integers into a stars-and-bars problem by introducing a slack variable.
Let $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ where $x, y, z$ are positive integers. Then $\vec{r} \cdot \vec{a} = x + y + z \leq 12$. We need to count positive integer solutions to $x + y + z \leq 12$, which is equivalent to $x + y + z + w = 12$ where $x, y, z \geq 1$ and $w \geq 0$. Substituting $x' = x-1, y' = y-1, z' = z-1$ where $x', y', z' \geq 0$, we get $x' + y' + z' + w = 9$. The number of non-negative integer solutions is $\binom{9+3}{3} = \binom{12}{3}$. This counts all valid vectors including the case where the sum equals 12.
Correct Answer: 2,3