From the word COMBINE, how many different 7 letter words are possible :

- with I adjacent to N?
- with alternating vowels and consonants?
-if a 5 letter word instead of 7 letter word is made from COMBINE, how many probability are that has exactly 3 consonants?

-with I adjacent to N

treat "IN" as a single element, so you now have to arrange 6 different elements, which is 6!
but it could be IN or NI,
so the number of arrangements is 2(6!) = 1440

- alternating vowels and consonants

Since you have 3 vowels and 4 consonants, you must start and end with a consonant to look like this
CVCVCVC
number of ways
= 4x3x3x2x2x1x1 , the bold represents the vowels
= 144

Give the last one a try. Let me know what you got with your reasoning.

To find the number of different 7-letter words possible from the word "COMBINE" with specific conditions, we can use the principles of permutation and combination.

1. With "I" adjacent to "N":
To find the number of words where the letter "I" is adjacent to the letter "N," we can treat "IN" as one unit. So, we have six units (C, O, M, B, NE, and the unit IN), which can be rearranged in 6! = 720 ways.

2. With alternating vowels and consonants:
To find the number of words where the vowels (O, I, and E) alternate with consonants (C, M, B, and N), we need to consider placing the vowels and consonants in a specific order. Since there are three vowels and four consonants, the pattern will be VCVCVCV or CVCVCVC. We will calculate the possibilities for both patterns and then add them together.

Pattern VCVCVCV:
- Vowels (O, I, E) can be arranged in 3! = 6 ways,
- Consonants (C, M, B, N) can be arranged in 4! = 24 ways.

Therefore, the total number of words with the VCVCVCV pattern is 6 x 24 = 144.

Pattern CVCVCVC:
- Consonants (C, M, B, N) can be arranged in 4! = 24 ways,
- Vowels (O, I, E) can be arranged in 3! = 6 ways.

Therefore, the total number of words with the CVCVCVC pattern is 24 x 6 = 144.

Adding the possibilities for both patterns, we have a total of 144 + 144 = 288 words with alternating vowels and consonants.

3. If a 5-letter word is made from "COMBINE" (instead of 7 letters) and exactly 3 consonants are required:
To find the number of words with exactly 3 consonants and 2 vowels, we need to consider the positions of the consonants and vowels within the word.

Number of ways to choose 3 consonants: C(4, 3) = 4 (since there are 4 consonants in "COMBINE").
Number of ways to choose 2 vowels: C(3, 2) = 3 (since there are 3 vowels in "COMBINE").
Number of ways to arrange the consonants: 3! = 6 (since we selected 3 consonants).

Number of ways to arrange the vowels: 2! = 2 (since we selected 2 vowels).

Therefore, the total number of 5-letter words with exactly 3 consonants is 4 x 3 x 6 x 2 = 144.

Note: In these calculations, I'm assuming repetition of letters is not allowed unless explicitly mentioned.