How many words can be made out of the letters of the word INDEPENDENCE, in which vowels always are together.
A) 16800
B) 16630
C) 1663200
D) None of these

Hint:
We can find the number of vowels and consonants. Then we can consider all the vowels as one letter and find the number of ways of arranging the other letters with the vowels. Then we can find the number of ways the vowels can be arranged themselves. Then we get the total number of words by taking the product of these.

Complete step by step solution:
We have the word INDEPENDENCE.
It has 12 letters, out of which 5 are vowels and 7 are consonants.
As we need to keep the vowels together always, we can consider the 5 vowels as one letter.
Then we need to arrange 8 letters. Out which D is occurring twice and N is occurring thrice.
So, the number of ways of arranging the consonants is given by, $\dfrac{{8!}}{{3! \times 2!}}$
On expanding the factorial, we get, \[\dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3!}}{{3! \times 2!}}\]
On simplification, \[\dfrac{{8 \times 7 \times 6 \times 5 \times 4}}{2}\]
=3360
Now the vowels can also be rearranged themselves. Out of the 5 vowels, 4 are the same.
So, the ways of arranging the vowels is given by $\dfrac{{5!}}{{4!}}$
On simplification, we get, $\dfrac{{5 \times 4!}}{{4!}} = 5$
So, the vowels can be arranged in 5 ways.
The number of words that can be formed such that vowels are always together is given by the product of the number of ways of arranging the letters with all the vowels together and the number of ways of arranging the vowels.
 $ \Rightarrow 5 \times 3360 = 16800$
So, the number of words can be made out of the letters of the word INDEPENDENCE, in which vowels always are together is 16800
Therefore, the required answer is 16800.

So, the correct answer is option A.

Note:
We used the concept of permutations to solve this problem. While calculating the permutations of arranging the letters we must divide the permutations with the factorial of the number of times all the letters are repeating. We must also take care that the vowels taken together can also be arranged in different ways. Alternate method to find the number of ways of arranging the vowels is by, there is 1 I and 4 Es. Then I can occupy any of the 5 places and E will occupy the other places. So, the number of ways of arranging the vowels is 5.