Animated Solution for Mathematics - Permutations and Combinations: The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is

First, let us dissect our subject. The word is $INDEPENDENCE$. If we count them carefully, we find a total of $12$ letters.

However, this is not a collection of unique characters. We have a hidden rhythm of repetitions: $I$: $1$ time $N$: $3$ times $D$: $2$ times $E$: $4$ times $P$: $1$ time $C$: $1$ time

This repetition is the heartbeat of the problem. If we ignore it, our answer will be inflated, a ghost of the true value.

The constraint is clear: all vowels must occur together. Imagine the vowels $I,E,E,E,E$ are a group of friends who refuse to be separated.

To handle this, we use the 'String Method.' We tie these five vowels into a single, unbreakable block. Now, instead of $12$ individual letters, we have $7$ consonants ($N,N,N,D,D,P,C$) and $1$ 'vowel-block.'

This gives us $8$ objects to arrange in total.

Now, we arrange these $8$ objects. Because we have three $N$'s and two $D$'s, we must divide by the permutations of these identical items to account for the repetitions.

The formula for our external arrangement becomes:

Calculating this, we know $8!=40320$, $3!=6$, and $2!=2$. Thus:

This is the number of ways to place our 'vowel-block' among the consonants.

Inside that 'vowel-block,' the five vowels ($I,E,E,E,E$) are not static. They can rearrange themselves in $5!$ ways.

However, because we have four identical $E$'s, we must divide by $4!$ to avoid overcounting. The internal arrangement is:

This is the elegant simplicity of the internal shuffle. There are only $5$ distinct ways the vowels can sit within their block: the $I$ can be in the first, second, third, fourth, or fifth position.

To find the total number of arrangements, we apply the Fundamental Principle of Counting. We multiply the external possibilities by the internal possibilities:

There it is. The final answer is $16800$.

It is not just a number; it is the result of logical discipline. You have successfully navigated the constraints, respected the repetitions, and arrived at the truth.