Question

Number of Permutation of the Word EQUATIONS
In how many ways can the letters of the word 'EQUATIONS' be arranged if the vowels will not change their order?

Answer 1

Letters:   E U  A I O       N S Q T  :  nine of them.  all letters are different.

Permutations:  with vowels not changing the order.

Place the vowels in the order    E   U  A   I   O.   There are six gaps in total to the left, center and to the right of these 5 letters.  We need to divide the 4 consonants among these gaps, such that we can fill a gap with 0, 1, 2, 3, or 4 consonants.
  
To solve this we use the polynomial expansion method.
 
We need to find the coefficient of x^4  in the expansion of   (1+x+x^2+x^3+x^4)^6 .
We simplify the polynomial and ignore powers of x more than 4.
(1+x+x^2+x^3+x^4)^6  =  [(1 - x^5)/(1-x)]^6  =  [1 - x^5]^6 /(1-x)^6
coefficient of x^4 in  (1-x)^-6   let n = 6
coefficient of   x^r  in (1-x)^-n   is     (n+r-1) C r  

So the  answer = (6+4-1)C4 = 9C4 = 9*8*7*6/24 = 126