Solution: There are 11 integers (0â10), of which 6 are even (0, 2, 4, 6, 8, 10). The number of ways to choose 4 distinct even integers is $inom{6}{4}$. The total number of ways to choose 4 distinct integers is $inom{11}{4}$. The probability is $ rac{inom{6}{4}}{inom{11}{4}} = rac{15}{330} = rac{1}{22}$. $oxed{\dfrac{1}{22}}$
["Understanding Combinatorial Probability: Choosing 4 Distinct Even Integers from 11 Total Integers", "In probability and combinatorics, calculating the chances of selecting specific elements from a set is fundamental. A classic example involves choosing integers from a defined group — both in simplicity and educational clarity. Let’s explore a well-defined problem:", "Problem Setup\nWe begin with 11 integers: ( {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ). Among these, exactly 6 are even: ( {0, 2, 4, 6, 8, 10} ), and 5 are odd.", "Our task is to compute the probability of selecting 4 distinct even integers when randomly choosing any 4 distinct integers from the full set.", "---", "### Step 1: Total Number of Ways to Choose 4 Distinct Integers", "From 11 integers, the total number of 4-element combinations is given by the binomial coefficient:", "[\n\binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11 \ imes 10 \ imes 9 \ imes 8}{4 \ imes 3 \ imes 2 \ imes 1} = 330\n]", "This is the total pool of possible selections.", "---", "### Step 2: Number of Favorable Outcomes", "We want to count how many ways to choose 4 distinct integers all of which are even.", "Since there are 6 even integers, the number of ways to select 4 of them is:", "[\n\binom{6}{4} = \frac{6!}{4! \ imes 2!} = \frac{6 \ imes 5}{2 \ imes 1} = 15\n]", "These combinations consist only of even numbers — satisfying our condition.", "---", "### Step 3: Calculate the Probability", "Probability is the ratio of favorable outcomes to total possible outcomes:", "[\nP = \frac{\binom{6}{4}}{\binom{11}{4}} = \frac{15}{330} = \frac{1}{22}\n]", "This elegant result—a simple fraction of ( \dfrac{1}{22} )—emerges naturally from basic combinatorics.", "---", "### Why This Matters", "This problem illustrates core principles of combinatorial probability:\n- Finite population selection using binomial coefficients.\n- Conditional counting by restricting choices to subsets with defined properties.\n- Simplifying complex probabilities through basic factorials and combinations.", "Understanding these foundations empowers you to tackle more complex real-world problems—from lottery odds to statistical modeling.", "---", "Final Takeaway\nThe probability of randomly selecting 4 distinct even integers from the set ( {0, 1, \dots, 10} ) is:", "[\n\boxed{\dfrac{1}{22}}\n]", "A small but powerful example of how combinatorics brings clarity and precision to randomness."]
