Number of Words with Vowels Always Together in "INDEPENDENCE"

To find the number of words that can be made from the letters of the word "INDEPENDENCE" in which vowels always come together, we need to follow these steps:

Step 1: Identify the Vowels

First, let's identify the vowels in the word "INDEPENDENCE." The vowels in this word are I, E, E, and E.

Step 2: Group the Vowels

Next, let's group the vowels together as per the given condition. In this case, we have three E's that need to be together. Therefore, we can consider them as a single unit.

So, we can treat the vowels as one entity, "EEE," and consider it as a single letter while arranging the letters.

Step 3: Find the Number of Arrangements

Now, we have the letters "NNDDPNNC" along with the grouped vowels "EEE." We need to find the number of arrangements possible with these letters.

To calculate the number of arrangements, we'll use the concept of permutations. The formula for permutations is:

n! / (n1! * n2! * n3! * ... * nk!)

Where n is the total number of objects, and n1, n2, n3, ..., nk are the number of repetitions of each object.

In our case, we have 10 letters in total: 3 N's, 2 D's, 2 E's, 1 I, 1 P, and 1 C.

So, the number of arrangements would be:

10! / (3! * 2! * 2! * 1! * 1! * 1!) = 10! / (3! * 2! * 2!)

Step 4: Calculate the Number of Arrangements

Now, let's calculate the number of arrangements using the formula mentioned above:

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2

Therefore, the number of arrangements is:

3,628,800 / (6 * 2 * 2) = 3,628,800 / 24 = 151,200

Step 5: Final Result

So, there are 151,200 words that can be made out of the letters of the word "INDEPENDENCE" in which the vowels always come together.

Summary:

Using the concept of permutations, we determined that there are 151,200 words that can be formed from the letters of the word "INDEPENDENCE," ensuring that the vowels always come together.