# Thread: [SOLVED] Permutations no two objects together arrangment..

1. ## [SOLVED] Permutations no two objects together arrangment..

The letter of the word CONSTANTINOPLE are written on 14cards, one on each card. the cards are shuffled and then arranged in a straight line.

(d) How many arrangements are there when no two vowels are next to each other?

There are 5 vowels here as I see it, A, I, E, 2 x O.

If we let vowels = V and consonants = C, then an arrangement like this can happen:

We have (14 - 5) = 9 consonants

- C1 - C2 - C3 - C4 - C5 - C6 - C7 - C8 - C9 -

Now, we have 10 places to input the first vowel, 9 for 2nd, 8 for 3rd, 7 for 4th, 6 for 5th ...

I am not sure how to solve it next. I did (9! / (3! 2!)) x 10 x 9 x 8 x 7 x 6 but that doesn't work and I am not sure what am I doing wrong.

2. ## Arrangements with repeated letters

Hello struck
Originally Posted by struck
The letter of the word CONSTANTINOPLE are written on 14cards, one on each card. the cards are shuffled and then arranged in a straight line.

(d) How many arrangements are there when no two vowels are next to each other?

There are 5 vowels here as I see it, A, I, E, 2 x O.

If we let vowels = V and consonants = C, then an arrangement like this can happen:

We have (14 - 5) = 9 consonants

- C1 - C2 - C3 - C4 - C5 - C6 - C7 - C8 - C9 -

Now, we have 10 places to input the first vowel, 9 for 2nd, 8 for 3rd, 7 for 4th, 6 for 5th ...

I am not sure how to solve it next. I did (9! / (3! 2!)) x 10 x 9 x 8 x 7 x 6 but that doesn't work and I am not sure what am I doing wrong.
Your reasoning is OK as far as you've gone. The consonants, as you say, can be arranged in $\frac{9!}{3!2!}$ ways, and there are then 10 slots for the vowels. So we next choose 5 slots from these 10 in $^{10}C_5$ ways, and then place the vowels in them in $\frac{5!}{2!}$ ways.

Total: $\frac{9!}{3!2!}\times \frac{10!}{5!5!} \times \frac{5!}{2!} = \frac{9!10!}{3!2!5!2!} = 457,228,800$ ways.

How does that seem?

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# how many arrangements are there where no two vowels are next to each other

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