### Related Formula Probability of an event P(E') = 1 - P(E)$P(E') = 1 - P(E)$, where E$E$ is the complementary event. ### Core Logic The word GARDEN contains 6 distinct letters: G, A, R, D, E, N. Total number of permutations (words in set S) = 6! = 720$6! = 720$. The vowels present are A and E. In any random arrangement of these letters, there are only 2 possible mutual relative arrangements for the vowels: 1) A appears before E (alphabetical order) 2) E appears before A By symmetry, both relative arrangements are equally probable. ### Step 1: Calculate Probabilities Probability that vowels are in alphabetical order = frac12$\frac{1}{2}$. Therefore, the probability that the selected word will NOT have vowels in alphabetical order is: 1 - frac12 = frac12$1 - \frac{1}{2} = \frac{1}{2}$ ### Pattern Recognition Symmetry Shortcut: For any k$k$ distinct specific objects inside an arrangement of distinct items, the number of ways they can be sorted in a unique relative order is exactly 1/k!$1/k!$ of the total arrangements. Here k=2$k=2$ vowels, so probability of alphabetical order is 1/2! = 1/2$1/2! = 1/2$. Not alphabetical is 1 - 1/2 = 1/2$1 - 1/2 = 1/2$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Permutations and Combinations Class 12 Mathematics: Probability