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

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" – ReverseFlow Nov 20 '16 at 14:36
• @Kiran You answer is right and their answer is wrong. I have added my explanation below. – user940 Nov 20 '16 at 15:12
• Checked with Python, the answer is indeed $90720$, deleted mine. – barak manos Nov 20 '16 at 15:13

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 $105840-15120=90720.$
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 $$181440 - 75600 - 15120 = 90720$$