In how many different ways can the letters of the word CHIMPANZEE be arranged so that all the vowels

M4maths Previous Puzzles

13 Sep 2018 Arrangement

In how many different ways can the letters of the word CHIMPANZEE be arranged so that all the vowels never come together?

OPtion

1) 907200
2) 1814400
3) 1753920
4) 1653920
5) 1853920
6) 1253920
7) 1553920
8) 1743920
9) 1763920
10) None of these

Solution

Let's first find out the number of arrangements without any restrictions. The word 'CHIMPANZEE' has 10 letters where E occurs 2 times.
Therefore, number of ways in which the letters of the word 'CHIMPANZEE' can be arranged without any restrictions = 10!/2! = 1814400

Now let's find out the number of arrangements where all the vowels come together. The word 'CHIMPANZEE' has 10 letters where 4 are vowels (I, A, E, E).
Vowels must come together. Therefore, group these vowels and consider it as a single letter.
i.e., CHMPNZ, (IAEE)
Thus we have total 7 different letters.
Number of ways to arrange these 7 letters = 7! = 5040


In the 4 vowels, E occurs 2 times.
Number of ways to arrange these 4 vowels among themselves = 4!/2!=12

Therefore, number of ways in which the letters of the word 'CHIMPANZEE' can be arranged where all the vowels come together =5040×12=60480



Number of ways in which the letters of the word 'CHIMPANZEE' can be arranged where all the vowels never come together= 1814400 − 60480
= 1753920
Option 3)

Correct Option: 3 Best Solution (3)

Kariamal   6 years AGO

chmpnz (iaee)
Vowels come together=7!*4!/2!=60480
Total=10!/2!=1814400
Vowels not together=1814400-60480
=1753920

Like? Yes (2) | No   

SAYAN CHATTERJEE   6 years AGO

Total no of arrangements is 10!/2
Among them AIEE comes together is 7!*4!/2! cases..
So, the no. of desired cases are 10!/2!- 7!*4!/2! = 1753920.
Option 3.

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Supriya Chavan   6 years AGO

Vowels never together = total ways - vowels together
= 10!/2 - (7!4!)/2!
= 1753920

Like? Yes | No