Latest CAT Aptitude Question SOLUTION: In how many ways can the letter of the word RAINBOW be arranged so that only 2 vowels always remain together?

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In how many ways can the letter of the word RAINBOW be arranged so that only 2 vowels always remain together?

Asked In CAT Thumma Mahender (11 years ago)

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Vinay Sipani   (11 years ago)

3C2 *2! * 6! = 4320 ways

Selecting two vowels out of 3 => 3C2
Arrangement of two vowels => 2!

Considering two vowels as a single letter,number of letters = 6
=> number of ways of arrangement = 3C2 * 2! * 6! = 4320 ways

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Aman  (11 years ago)

@ Vinay,

Pls check.

In 6! cases, there will be some cases when group of two selected vowels will be placed next to third vowel which is not allowed.

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Vinay Sipani   (11 years ago)

Thank you Aman

Answer is 2880

number of ways = 4320 - number of ways so that the third vowel is next to two vowels
= 4320 - (3C2 * 2! * 2! * 5!)
= 4320 - 1440
= 2880

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Inder  (11 years ago)

Number of ways the letter of the word RAINBOW can be arranged so that only 2 vowels always remain together = 3*(6!*2 - 5!*4) = 3*960 = 2880

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vinisha  (11 years ago)

@ VINAY SIPANI

while calculating number of ways so that the third vowel is next to two vowels
how did you get (3C2 * 2! * 2! * 5!) can you explain me this part...
because i got (3C2 * 2! * 5!)

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SATISH VARADA  (11 years ago)

2600
AI,R,N,B,O,W = 6!*2
+
AO,R,I,N,B,W = 6!*2
+
IO,R,B,N,A,W = 6!*2
-
AOI,R,N,B,W = 3!*5!
=2600

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sravani  (9 years ago)

I think 4320 ways because...
2 out of 3 vowels can be arranged in 3p2 ways
and the total letters with 2 vowels as an entity are 6 which can be arranged in 6! ways..
so, 6!*3p2=4320.

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DEEPIKA MONINGI  (8 years ago)

Taking Two Vowels As One Entity , We Get 3P2 *6! = 4320 Ways.
Number Of Ways So That The Third Vowel Is Next To Two Vowels :

VV*CCCC - 1 x 4!
*VV*CCC - 2 x 4!
C*VV*CC - 2 x 4!
CC*VV*C - 2 x 4!
CCC*VV* - 2 x 4!
CCCC*VV - 1 x 4!
so it sums up to 240
so required no of ways is 4320-240 = 4080

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