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 : [2025]

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Solution

Q.
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 : [2025]

1 $\frac{1}{3}$

2 $\frac{2}{3}$

3 $\frac{1}{4}$

4 $\frac{1}{2}$

Ans.
(4)

We have, A E G R D N

Total cases = 6!

Number of cases when A and E are in order = ${}^{6}C_{2} \times 4!$

Probability, when A and E are in order = $\frac{{}^{6}C_{2} \times 4!}{6!} = \frac{1}{2}$

Probability when A and E are not in order = $1 - \frac{1}{2} = \frac{1}{2}$

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