Q.The number of different words that can be formed from the letters of the word INTERMEDIATE such that two vowels never come together is ______.

The key idea is to use permutations with repetitions, applying the "gap method" to ensure no two vowels come together.

1. Identify Vowels and Consonants:

   The word INTERMEDIATE has 12 letters.

   Vowels: I (2 times), E (3 times), A (1 time) $\implies$ 6 vowels.

   Consonants: N (1 time), T (2 times), R (1 time), M (1 time), D (1 time) $\implies$ 6 consonants.

2. Arrange the Consonants:

   The 6 consonants (N, T, R, M, D, T) can be arranged in $\frac{6!}{2!}$ ways, as 'T' is repeated twice.

   Number of ways to arrange consonants $= \frac{720}{2} = 360$.

3. Place the Vowels in Gaps:

   Arranging 6 consonants creates $6+1=7$ possible positions (gaps) where the vowels can be placed so that no two vowels are adjacent.

   We must choose 6 of these 7 positions for the 6 vowels: $\binom{7}{6} = 7$ ways.

   The 6 vowels (I, I, E, E, E, A) can be arranged in these 6 chosen positions in $\frac{6!}{2!3!1!}$ ways, considering repetitions.

   Number of ways to arrange vowels $= \frac{720}{2 \times 6 \times 1} = \frac{720}{12} = 60$. …

Let's break down this problem by first understanding the constraint: "two vowels never come together". This means that between any two vowels, there must be at least one consonant. The most effective strategy for such problems is to arrange the "restricted" items (vowels, in this case) in the spaces created by the "unrestricted" items (consonants).

First, we need to analyze the letters in the given word, "INTERMEDIATE", and categorize them into vowels and consonants, noting any repetitions.

The word INTERMEDIATE has 12 letters.

Let's list them and count their frequencies:

• I: 2
• N: 1
• T: 2
• E: 3
• R: 1
• M: 1
• D: 1
• A: 1

Now, let's separate them into vowels and consonants:

Vowels (V): A, E, E, E, I, I

• A: 1
• E: 3
• I: 2 Total Vowels = 6

Consonants (C): N, T, R, M, D, T

• D: 1
• M: 1
• N: 1
• R: 1
• T: 2 Total Consonants = 6

We have 6 vowels and 6 consonants.

Here's the step-by-step approach:

1. Arrange the Consonants: We have 6 consonants: N, T, R, M, D, T. Among these, the letter 'T' is repeated 2 times. The number of distinct ways to arrange these consonants is given by the permutation formula for items with repetitions:

There are 360 distinct ways to arrange the consonants.

2. Create Spaces for Vowels:

When we arrange 6 consonants, they create $6+1=7$ possible spaces where vowels can be placed. Placing vowels in these spaces ensures that no two vowels will be adjacent.

Imagine the consonants (C) arranged, creating slots (represented by underscores):

_ C _ C _ C _ C _ C _ C _

There are 7 such spaces.

3. Place the Vowels in the Spaces: We have 6 vowels: A, E, E, E, I, I. We need to place these 6 vowels into 6 of the 7 available spaces.
   • Choose the spaces: First, we select 6 spaces out of the 7 available spaces. The number of ways to do this is given by the combination formula:

    There are 7 ways to choose the 6 spaces. …