### Related Formula P(E) = fractextNumber of favorable outcomestextTotal number of possible outcomes$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ ### Core Logic Total possible selections for (x, y)$(x, y)$ with replacement from 0, 1, dots, 10${0, 1, \dots, 10}$ is 11 times 11 = 121$11 \times 11 = 121$. We need pairs (x, y)$(x, y)$ such that |x - y| > 5$|x - y| > 5$, which means x - y > 5$x - y > 5$ or y - x > 5$y - x > 5$. ### Step 1: Counting Favorable Cases Assume x < y$x < y$, so we need y - x geq 6$y - x \geq 6$. If x = 0 Rightarrow y in \6, 7, 8, 9, 10\$x = 0 \Rightarrow y \in \{6, 7, 8, 9, 10\}$ (5 ways) If x = 1 Rightarrow y in \7, 8, 9, 10\$x = 1 \Rightarrow y \in \{7, 8, 9, 10\}$ (4 ways) If x = 2 Rightarrow y in \8, 9, 10\$x = 2 \Rightarrow y \in \{8, 9, 10\}$ (3 ways) If x = 3 Rightarrow y in \9, 10\$x = 3 \Rightarrow y \in \{9, 10\}$ (2 ways) If x = 4 Rightarrow y in \10\$x = 4 \Rightarrow y \in \{10\}$ (1 way) If x ge 5$x \ge 5$, there are no possible values for y$y$ strictly greater than x$x$ satisfying the condition. ### Step 2: Total Probability The number of cases for y > x$y > x$ is 5 + 4 + 3 + 2 + 1 = 15$5 + 4 + 3 + 2 + 1 = 15$. By symmetry, the number of cases for x > y$x > y$ is also 15$15$. Total favorable cases = 15 times 2 = 30$= 15 \times 2 = 30$. Required probability = frac30121$= \frac{30}{121}$. ### Pattern Recognition Absolute difference conditions |x - y| > k$|x - y| > k$ on discrete sets cleanly split into symmetric additive series 1+2+...+n$1+2+...+n$. Calculate one half and multiply by 2. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Maths: Probability