Question

Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

• A. \( \frac{1}{4} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{2} \)

Hint:

For problems involving probability, it is often useful to calculate the complementary event and subtract from 1.

Answer 1

The Correct Option is D

The word GARDEN comprises the letters G, A, R, D, E, N. The vowels within this set are A and E. The subsequent steps outline the calculation of the probability that a randomly selected arrangement of these letters will NOT have the vowels in alphabetical order.
Step 1: Total possible arrangements. 
With 6 distinct letters, the total number of unique arrangements is calculated as: \[ {Total arrangements} = 6! = 720 \] 
Step 2: Arrangements with vowels in alphabetical order. 
To ensure the vowels A and E are in alphabetical order (A preceding E), we consider all possible positions for A and E. The number of arrangements where A appears before E is determined by: \[ {Favorable cases} = \binom{6}{2} \cdot 4! = 15 \cdot 24 = 360 \] 
Step 3: Probability calculation. 
The probability of selecting an arrangement where the vowels are in alphabetical order is: \[ P = \frac{360}{720} = \frac{1}{2} \] Consequently, the probability of selecting an arrangement where the vowels are NOT in alphabetical order is: \[ P({Not in order}) = 1 - \frac{1}{2} = \frac{1}{2} \]