# If we rearrange the letters in the word "strange", what is the ratio of rearrangements where the vowels are not together to the rearrangements where they are together?

How many of those ways will have the vowels together?

There are 2 vowels and they can be arranged 2 different ways: AE and EA.

There are 6 different places in the word these two vowels can be: positions 1, 2; 2, 3; 3, 4...all the way to 6, 7.

And so there are:

This means there are:

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To find the ratio of rearrangements where the vowels are not together to the rearrangements where they are together, we first need to determine the total number of rearrangements of the letters in the word "strange."

The word "strange" has 7 letters, with 2 vowels (a and e) and 5 consonants (s, t, r, n, g).

Total rearrangements = 7! / (2! * 5!) = 5040 / 240 = 21.

To calculate the number of rearrangements where the vowels are together, we treat the vowels (a and e) as one entity. This gives us a total of 6 entities to arrange: (strng), (st), (r), (n), (g), (ae).

Arrangements where the vowels are together = 6! * 2! = 720 * 2 = 1440.

Now, to find the rearrangements where the vowels are not together, we subtract the arrangements where the vowels are together from the total arrangements.

Rearrangements where vowels are not together = Total rearrangements - Arrangements where vowels are together = 21 - 1440 = 7.

Therefore, the ratio of rearrangements where the vowels are not together to the rearrangements where they are together is 7:1440, which can be simplified to 1:205.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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