How many different words can be formed with all the letters of the word "INTERNET" if each word is to begin with vowel? Plead help me to figure out the solution! According to the book it's answer is 1890. Please help me!

The number of anagrams of the word INTERNET is $$\frac{8!}{2!\cdot 2! \cdot 2!}=5040$$ because there are 8 letters with 2 Ns, 2 Es and 2 Ts.

How many of them begin with a vowel?

Now the first letter can be chosen in $3$ ways and the remaining places can be filled in $(8-1)!$ ways. Hence, in the above fraction, we replace $8!$ with $3\cdot 7!$: $$\frac{3\cdot 7!}{2!\cdot 2! \cdot 2!}=1890.$$