Here are two additional methods:
Method 1: We use symmetry.
The word $ENGINEER$ has eight letters, of which three are $E$s, two are $N$s, one is a $G$, one is an $I$, and one is an $R$. We can choose three of the positions for the $E$s, two of the remaining five positions for the $N$s, and arrange the remaining three distinct letters in the three remaining spaces in
By symmetry, one fourth of these arrangements will have an $I$ in the second position among the four vowels. Hence, the number of admissible arrangements is
Method 2: We place the consonants, then the vowels.
We choose two of the eight positions for the two $N$s, one of the remaining six positions for the $G$, and one of the remaining five positions for the $R$. There is only one way to arrange the vowels in the remaining four positions so that they appear in the same order that they do in the word $ENGINEER$. Hence, the number of admissible arrangements is