Arrangements of ten letters

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Question from Mustafa, a student:

Hello there,
I was doing a Mathematics Paper when I stumbled upon this.
I figured I must have solved these kind of problems many times before
but I was hopelessly stuck this time. Hence this query to you. A solution
and explanation would be much appreciated. Thank You.

The problem is:

In how many ways can the ten letters of the word GELATINOUS be arranged
in a line so that the vowels are in alphabetical order (not necessarily together)
when read from left to right?

PS- the correct answer seems to be 30240 according to the book, but I cannot
figure it out.

Mustafa,

You can line up 10 distinct objects in 10! ways. In your example though 5 of the objects have to be in a specific order; 5 such objects could be lined up in 5! ways with no restrictions. Thus your problem can be solved in 10!/5! ways = 30240 ways.

Penny

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