Here, it is clear that 3 things are already selected and we need to choose (r – 3) things from the remaining (n – 3) things.

Let us find the no. of ways of choosing (r – 3) things.

Number of ways = (No. of ways of choosing (r – 3) things from remaining (n – 3) things)

= ^{n – 3}C_{r – 3}

Now we need to find the no. of permutations than can be formed using 3 things which are together. So, the total no. of words that can be formed is 3!

Now let us assume the together things as a single thing this gives us total (r – 2) things which were present now. So, the total no. of words that can be formed is (r – 2)!

Total number of words formed is:

^{n – 3}C_{r – 3} × 3! × (r – 2)!

The no. of permutations that can be formed by r things which are chosen from n things in which 3 things are always together is ^{n – 3}C_{r – 3} × 3! × (r – 2)!

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?

How many words, with or without meaning can be formed from the letters of the word ‘MONDAY’, assuming that no letter is repeated, if

**(i)** 4 letters are used at a time

**(ii)** all letters are used at a time

**(iii)** all letters are used but first letter is a vowel ?

There are 10 persons named P_{1}, P_{2}, P_{3} …, P_{10}. Out of 10 persons, 5 persons are to be arranged in a line such that is each arrangement P_{1} must occur whereas P_{4} and P_{5} do not occur. Find the number of such possible arrangements.

How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?

How many triangles can be obtained by joining 12 points, five of which are collinear?