Illustration 1:

(a)       How many anagrams can be made by using the letters of the word HINDUSTAN?

(b)       How many of these anagrams begin and end with a vowel.

(c)       In how many of these anagrams, all the vowels come together.

(d)       In how many of these anagrams, none of the vowels come together.

(e)       In how many of these anagrams, do the vowels and the consonants occupy the same relative positions as in HINDUSTAN?

Solution: Illustration 2:

How many 3 digit numbers can be formed using the digits 0, 1, 2,3,4,5 so that

(a)       Digits may not be repeated

(b)       Digits may be repeated

Solution:

(a)       Let the 3-digit number be XYZ

Position (X) can be filled by 1, 2,3,4,5 but not 0. So it can be filled in 5 ways.

Position (Y) can be filled in 5 ways again. (Since 0 can be placed in this postion).

Position (Z) can be filled in 4 ways.

Hence, by the fundamental principle of counting, total number of ways is

5 x 5 x 4 = 100 ways.

(b)       Let the 3 digit number be XYZ

Position (X) can be filled in 5 ways

Position (Y) can be filled in 6ways.

Position (Z) can be filled in 6 ways.

Hence by the fundamental principle of counting, total number of ways is

5 x 6 x 6 = 180.