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:

Solution

Maths sir · 4 steps

Only two vowels, so A comes before E in exactly half of all the words.

1. Count all words. GARDEN has 6 different letters, so n(S) = 6! = 720.
2. Pair up the words. The vowels are A and E. Swap the places of A and E in any word and you get another word in S. In one word of each pair A comes first, and in the other E comes first.

   n(A before E) = 6!/2! = 360

3. Take the complement. Words with the vowels not in alphabetical order are those with E before A, and there are 720 - 360 = 360 of them.
4. So. The probability is 360/720 = 1/2, which is option (b).

The idea to keep. If k distinct letters must appear in one fixed order, divide the total arrangements by k!. With two vowels that is half, so the opposite order is also half.

Answer from the NTA final answer key. Solution by PrepWiser.