How to find number of words made using letters of word 'EQUATION' if order of vowels do not change Find number of words made using letters of word 'EQUATION' if order of vowels do not change.
My attempt:-  since we do not have to change the order of the vowels  hence, 
             _E_U_A_I_O_
we have $6$ places to fill the remaining letters. Therefore total number of cases $= C(6,3) \times 3! + 6 \times 3!= 156$ 
But the answer is $336$.

There are simpler approaches.
Method 1: We place the consonants first.  
Since the word EQUATION has eight letters, there are eight positions to fill.  Therefore, there are eight ways to place the Q, seven ways to place the T, and six ways to place the N.  Since the vowels must appear in the original order, there is only one way to place the five vowels in the remaining five positions.  Hence, the number of arrangements of the letters of the word EQUATION in which the vowels appear in the original order is $8 \cdot 7 \cdot 6 = 336$.
Method 2:  We use symmetry.
Since the word EQUATION has eight distinct letters, there are $8!$ ways to arrange its letters.  Within a given arrangement, the five distinct vowels can be permuted among themselves in $5!$ ways.  Of these $5!$ ways of arranging the vowels, only one leaves the vowels in the original order.  Hence, the number of admissible arrangements of the letters is 
$$\frac{8!}{5!} = \frac{8 \cdot 7 \cdot 6 \cdot 5!}{5!} = 8 \cdot 7 \cdot 6 = 336$$
