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:
Show Hint
- \( \frac{1}{4} \)
- \( \frac{2}{3} \)
- \( \frac{1}{3} \)
- \( \frac{1}{2} \)
The Correct Option is D
Solution and Explanation
We are given the word GARDEN, which consists of the following letters: G, A, R, D, E, N. Among these letters, the vowels are A and E. To find the probability that the selected word will NOT have vowels in alphabetical order, we proceed as follows:
Step 1: Total number of arrangements.
Since there are 6 distinct letters in the word GARDEN, the total number of ways to arrange these letters is: \[ {Total arrangements} = 6! = 720 \]
Step 2: Number of favorable cases (vowels in alphabetical order).
For the vowels A and E to be in alphabetical order, the positions of A and E must be such that A appears before E. The total number of ways to arrange the 6 letters such that A appears before E is: \[ {Favorable cases} = \binom{6}{2} \cdot 4! = 15 \cdot 24 = 360 \]
Step 3: Probability calculation.
The probability that the selected word will have vowels in alphabetical order is: \[ P = \frac{360}{720} = \frac{1}{2} \] Therefore, the probability that the selected word will NOT have vowels in alphabetical order is: \[ P({Not in order}) = 1 - \frac{1}{2} = \frac{1}{2} \]
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