This single question contains 80%-90% concept of permutation.
In how many ways the letters of the word R,N,A,I,B,W,O be arranged?
(1) How many words begin with N?
(2) How many words begin with N and ends with R?
(3) How many words are there in which A and W are at the end positions?
(4) How many words are there in which W and I are together?
(5) How many words are there in which A and B are never together?
(6) How many words are there in which vowels are never together?
(7) How many words are there in which vowels are always before consonants?
(8) How many words are there in which first and last letters are vowels?
(9) If arrangements formed are arranged in dictionary form, then what is the position of the word RAINBOW in that dictionary?
Total ways of arranging the letters = 7! = 5040 ways.
(1) When N is fixed at initial, then we have only 6 letters left to arrange the word. Total ways = 6! = 720 ways
(2) When now N is fixed at initial, and R is fixed at the end, we have only 5 letters left to arrange the word.
Total ways = 5! = 120
(3) When we have A and W at initial and end, we can arrange the word in 120 ways. In a similar way, 120 ways are there when W is at initial and A is at end. SO a total of 120+120 = 240 ways are there for this question.
(4) Assume W and I stick together as one letter, we have 6 letters for the arrangement. ( R,A,N,B,O,WI ) which can be arranged in 6! ways = 720 ways. But WI can also be mutually arrange in two ways (WI and IW). Total final ways of arranging the word = 6!*2! = 1440 ways.
(5) If A and B are together, then the number of ways of arranging = 1440 ways — – [see (4)’s explanation]
Out of total permutations of 5040 ways, 1440 ways are there in which I and O are together. Remaining will be the ones in which A and B are not together.
A and B not together = 5040-1440 = 3600 ways
(6) There vowels are there: A, I, O
Either vowel can initiate the word (like ARINBOW) or a consonant can initiate the word (like RAINBOW)
Either vowel can end the word (like RAINBWO) or a consonant can end the word (like RAINBOW)
For vowels to not come together, they must come in between consonants. When 4 consonants are arranged, this leaves five places for three vowels to fill
C = consonant
_ = vowel
Thus, 5 places can be filled by 3 vowels in = 5P3 = 5!/2! = 60 ways
And four places of consonants can be filled in = 4! = 24 ways
Total ways = 60*24 = 1440 ways
(7) Let’s combine vowels AIO as one letter and consonants RNBW as one letter, giving AIO, RNBW as two letters. These two letters can be arranged in 2 ways
i.e. AIORNBW; or RNBWAIO;
But vowels come before consonants. Thus, there is only one way of doing this.
And also, Vowels can mutually arrange in 3! = 6 ways
And consonants can mutually arrange in 4! = 24 ways
Thus, total ways = 1*6*24 = 144 ways
(8) 3 vowels are there. First and last (two places) can be filled by these 3 vowels in = 3P2 = 3!/1! = 6 ways
And remaining letters (4 consonants, 1 vowel = 5 letters) can be arranged in 5! ways
Total 6*5! = 720 ways
(9) In dictionary, A comes before B and B comes before C and so on. Let’s arrange the letters in word RAINBOW
Ascending order ==> ABINORW
If 1st letter = ‘A’, the number of words beginning with A = 6! (There are 6 letters left to arrange)
Similarly, number of words beginning with ‘B’ = 6!
With ‘I’ = 6!
With ‘N’ = 6!
With ‘O’ = 6!
With ‘RAB’ = 4! (When we begin with R, A will automatically come next to R in dictionary and B will come next to A. So, we’ll begin with RAB)
With ‘RAIB’ = 3! (When arrangements of RAB is finished, arrangements of RAI begins, and B comes next to RAI. So, RAIB)
The next letter with RAIN will be RAINBOW
RAINBOW will come after = 5*6! + 4! + 3! = 3630th places. I.e. RAINBOW will come at 3631’s place.
For more clarification, the letters arranged with RAIN as first letters will 3! = 6 ways = RAINBOW, RAINBWO, RAINOBW, RAINOWB, RAINWBO, RAINWOB
(Written by Atul)