How many words can be formed with the letters of the word 'BINOMIAL'? In how many words, vowels will occur together?

Answer

To solve the problem, we will break it down into two parts:

1. Finding the total number of words that can be formed with the letters of the word "BINOMIAL".
2. Finding the number of words in which the vowels occur together.

### Step 1: Total Number of Words

The word "BINOMIAL" consists of 8 letters: B, I, N, O, M, I, A, L.

- We note that the letter 'I' appears twice, while all other letters appear once.

To find the total number of distinct arrangements of the letters, we use the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

Where:
- \( n \) is the total number of letters,
- \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

Here, \( n = 8 \) (total letters), and the letter 'I' appears twice:

\[ \text{Total arrangements} = \frac{8!}{2!} \]

Calculating this:

\[ 8! = 40320 \]
\[ 2! = 2 \]
\[ \text{Total arrangements} = \frac{40320}{2} = 20160 \]

### Step 2: Number of Words with Vowels Together

The vowels in "BINOMIAL" are I, I, O, A. We treat these vowels as a single unit or block.

So, we have the following units to arrange:
- Vowel block (IIOA)
- B
- N
- M
- L

This gives us a total of 5 units: (IIOA), B, N, M, L.

Now, we will calculate the arrangements of these 5 units:

\[ \text{Arrangements of units} = 5! \]

Calculating this:

\[ 5! = 120 \]

Next, we need to arrange the vowels within the vowel block (IIOA). Since 'I' appears twice, we use the formula for permutations of multiset again:

\[ \text{Arrangements of vowels} = \frac{4!}{2!} \]

Calculating this:

\[ 4! = 24 \]
\[ 2! = 2 \]
\[ \text{Arrangements of vowels} = \frac{24}{2} = 12 \]

Now, we multiply the arrangements of the units by the arrangements of the vowels:

\[ \text{Total arrangements with vowels together} = 5! \times \frac{4!}{2!} = 120 \times 12 = 1440 \]

### Final Answers

1. The total number of words that can be formed with the letters of the word "BINOMIAL" is **20160**.
2. The number of words in which the vowels occur together is **1440**.