[Math] Arranging the letters of the word EDUCATION, so that the following conditions hold

combinationscombinatoricspermutations

How many ways to arrange the letters of the word EDUCATION, so that the following conditions hold:

  • The vowels occur in the same order
  • The consonants occur in the same order
  • no two consonants are next to each other

My approach:

Suppose I arrange the consonants and so it looks like
_D_C_T_N_
Now we need to arrange the vowels in the same order as they occur in the original word(they are being placed in the underscores)

Now, note that there are $5$ underscores and total no. of vowels $=5$

So, the problem boils down to the following:

Find all integer solutions to the equation:$$x_1+x_2+x_3+x_4+x_5=5,$$
where $x_2,x_3,x_4$ are not equal to $0$

I said that $x_2,x_3,x_4$ are not equal to $0$, because no two consonants are next to each other, but $x_1$ and $x_5$ can be $0$.

Now this is a modification of the famous Stars and Bars problem. How to go from here?

Best Answer

It is possible to use unmodified stars and bars by considering the vowels first:

_E_U_A_I_O_

where each underscore may hold at most one consonant. Immediately we see that the answer to the question is $\binom64=15$.

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