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# How many arrangements can be made by using the letters of the word GARDEN so that the vowels are in alphabetical order?

Last updated date: 17th Jun 2024
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Hint: We will look at the number of vowels in the word GARDEN and their alphabetical order. Then we will count the number of ways to arrange these vowels so that their alphabetical order is maintained. We will look at the remaining letters, which are the consonants of the word GARDEN. We will have to count the number of ways these letters can be arranged. Then, we will find the total number of possible arrangements of the letters of the word GARDEN which satisfy the given conditions. Here we arrange n elements at n places by $n!= n\times(n-1)\times (n-2).......3\times 2 \times 1$
$5+4+3+2+1=15$
Now, the remaining letter of the word GARDEN is the 4 consonants. Taking into account two slots for the vowels, these four consonants can be arranged in the remaining four slots in $4!$ ways.
Therefore, the total number of arrangements of all the letters of the word GARDEN such that the vowels are in alphabetical order is $15\times 4!=15\times 24=360$.