Permutation On Finding The Rank Of The Word As in a Dictionary

What is Permutation?

Permutation means arrangement, we can arrange letter, numbers, and things.

Types of permutation problems:
i) Letters Arrangement                        ii) Numbers Arrangement
   a.)Vowels Come Together                    a.)Repetitions are allowed
   b.)Vowels Not Come Together             b.)Repetitions are not allowed
   c.)Dictionary Arrangement
        1.)Ranking Method

Some of Basic Examples for Letters Arrangement problem:

1) In how many ways can letter of the word "Arun" can be arranged?

Solution:
 4P4 = 4! = 24                           ( npr = n!/(n-r)!)

Problems on Letters Arrangement:

1. In how many different ways the letters of the word TAMANNA be arranged?

Solution:

TAMANNA

Total letters = 7!
Repeating letters = AAA     NN    = 3! x 2!

                         =  7!/3!x2!
                         = 420

2.  In how many different ways the letter of the word BALAJI be arranged?

Solution:

BALAJI

Total letters = 6!
Repeating letters = AA  = 2!

                           = 6!/2!
                           = 360

Problems on Vowels Come Together 

1. How many arrangements can be made out of the letters of the word "PUDUCHERRY" taken all at the time, such that the vowels come together?

Solution:

PUDUCHERRY

Consider the consonant as separate letters and vowels as single letter

Arrangement of Vowels come together = PDCHRRYUUE

Total consonant+Vowels (single letter) = 7+1 = 8!/2! =   20160          
                         
( RR = 2!, UU is not considered as repeating letter because vowels are considered as single letter)

UUE = 3!/2! = 3         ( Internal arrangement in vowels can be taken place UUE, UEU, EUU)

arrangements of vowels come together =  20160 x 3
                                                               = 60480


2. How many arrangements can be made out of the letters of the word "PUDUCHERRY" taken all at the time, such that the vowels not come together?

Solution:

Vowels not come together = (total arragement) - (vowels come together)

         PUDUCHERRY

Total letter = 10!
Repeating letter = UU  RR = 2!x2!

total arrangement =  10!/2!x2!
                             =907200

vowels not come together = (907200) - ( 60480)
                                          = 846720