In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels

Question

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels

I solved and got answer as $90720$. But other sites are giving different answers. Please help to understand which is the right answer and why I am going wrong.

My Solution

Arrange 6 consonants $\dfrac{6!}{2!}$
Chose 2 slots from 7 positions $\dbinom{7}{2}$
Chose 1 slot for placing the 2 vowel group $\dbinom{2}{1}$
Arrange the vowels $3!$

Required number of ways:
$\dfrac{6!}{2!}\times \dbinom{7}{2}\times \dbinom{2}{1}\times 3!=90720$

Solution taken from http://www.sosmath.com/CBB/viewtopic.php?t=6126)

Solution taken from http://myassignmentpartners.com/2015/06/20/supplementary-3/

Can you explain your working. Just putting down your calculation doesn't tell us why you chose to do them. — Ian Miller, Nov 20 '16 at 14:30
@sorry, edited the calculation and added the details. pl help. — Kiran, Nov 20 '16 at 14:31
I will point out that the solution in the excerpt solves a different problem. Your problem asks for "exactly two consecutive vowels", the excerpt's solution allows 3 consecutive vowels as well. As it says at the end "with at least two adjacent vowel" — ReverseFlowControl, Nov 20 '16 at 14:36
$ReverseFlow, but in the first excerpt , they do 120960 - 15120 which count exactly two consequent vowels only. So this approach also must be right I guess. — Kiran, Nov 20 '16 at 14:37
Ah, I see. Yes, that strategy works for the counting. One sec. — ReverseFlowControl, Nov 20 '16 at 14:39
There are 6 consonants in which 2 are repeating. so number of arrangements possible using them alone is 6!/2!. this is what I meant — Kiran, Nov 20 '16 at 14:41
but then, the answer given in the first website (also given by @barak manos) must also be correct, right? why that is going wrong? — Kiran, Nov 20 '16 at 14:59
@Kiran You answer is right and their answer is wrong. I have added my explanation below. — , Nov 20 '16 at 15:12
Checked with Python, the answer is indeed $90720$, deleted mine. — barak manos, Nov 20 '16 at 15:13

score 3 · Accepted Answer · 2016-11-20T15:16:09.720

The number of arrangements with 3 consecutive vowels is correctly explained in the original post: the number is $15120$.

To find the number of arrangements with at least two consecutive vowels, we duct tape two of them together (as in the original post) and arrive at $120960$.

The problem with this calculation is that every arrangement with 3 consecutive vowels was double counted: once as $\overline{VV}V$ and again as $V\overline{VV}$. To compensate for this we must subtract $15120$. The correct number of arrangements with at least two consecutive vowels is $120960-15120=105840.$

Therefore, correct number of arrangements with exactly two consecutive vowels is $105840-15120=90720.$

yes, it is. I am able to understand now. thanks for your help, i spent over 6 hours for this problem only. I have to improve a lot :( — Kiran, Nov 20 '16 at 15:13

score 1 · Answer 2 · answered Nov 22 '16 at 06:05

The total number of ways of arranging the letters is $\frac{9!}{2!} = 181440$. Of these, let us count the cases where no two vowels are together. This is $$\frac{6!}{2!} \times \binom{7}{3}\times 3! = 75600$$ Again, the number of ways in which all vowels are together is 15120. Thus the number of ways in which exactly two vowels are together is $$181440 - 75600 - 15120 = 90720$$

In how many ways can the letters in WONDERING be arranged with exactly two consecutive vowels

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