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Permutations Questions Drill

by fangtray » Mon Sep 19, 2011 8:08 pm
Hello could someone please provide answers and explanations for the following?

In how many ways can the letters of the word "Computer" be arranged?

A. Without any restriction
B. M must always occur at the third place.
C. All vowels are together.
D. All the vowels are never together.
E. Vowels occupy the even positions.



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by amit2k9 » Mon Sep 19, 2011 8:21 pm
a 8 letters meaning 8!.

b 7*6*1*5*4*3*2*1 ways.

c vowels here = e,o,u. together as a unit = 3! ways to arrange among themselves.
remaining letters = 5 + 1 (vowel unit) = 6! ways to arrange.
thus number of ways = 3! * 6!

d total ways - number of ways in which vowels are together = 8! - 3!*6!

e 5*3(vowel-1)*4*2(vowel-2)*3*1(vowel-3)*2*1
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by fangtray » Mon Sep 19, 2011 8:41 pm
sorry could you put your answers b - e in simpler terms? i'm not sure why it works like that
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by Anurag@Gurome » Mon Sep 19, 2011 9:02 pm
fangtray wrote:Hello could someone please provide answers and explanations for the following?

In how many ways can the letters of the word "Computer" be arranged?

A. Without any restriction
B. M must always occur at the third place.
C. All vowels are together.
D. All the vowels are never together.
E. Vowels occupy the even positions.
Total number of letters in the word COMPUTER = 8

A. Without any restriction, the letters can be arranged in 8! = 40,320 ways.

B. No. of ways so that M must always occur at the third place (one place is fixed in this case) = 7! = 5,040 ways

C. No. of vowels in the word COMPUTER = 3 (O, U, E)
Consider the 3 vowels as 1 letter. Then there are 5 consonants, so it can be done in 6! = 720
Apart from this the 3 vowels can be arranged in 3! = 6 ways
No. of ways to arrange the letters in a way so that all vowels are together = = 720 * 6 = 4,320 ways

D. No. of ways to arrange the letters in a way so that the vowels are never together = 8! - 6! * 3! = 40,320 - 4,320 = 36,000 ways

E. There are 4 even positions that can be filled by the 3 vowels in 4 * 3 * 2 = 24 ways
Now 5 positions are left, which can be filled in 5! = 120 ways
No. of ways to arrange the letters in a way so that vowels occupy the even positions = 24 * 120 = 2,880 ways
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by fangtray » Tue Sep 20, 2011 2:27 am
Anurag@Gurome wrote:
fangtray wrote:Hello could someone please provide answers and explanations for the following?

In how many ways can the letters of the word "Computer" be arranged?

A. Without any restriction
B. M must always occur at the third place.
C. All vowels are together.
D. All the vowels are never together.
E. Vowels occupy the even positions.
Total number of letters in the word COMPUTER = 8

A. Without any restriction, the letters can be arranged in 8! = 40,320 ways.

B. No. of ways so that M must always occur at the third place (one place is fixed in this case) = 7! = 5,040 ways

C. No. of vowels in the word COMPUTER = 3 (O, U, E)
Consider the 3 vowels as 1 letter. Then there are 5 consonants, so it can be done in 6! = 720
Apart from this the 3 vowels can be arranged in 3! = 6 ways
No. of ways to arrange the letters in a way so that all vowels are together = = 720 * 6 = 4,320 ways

D. No. of ways to arrange the letters in a way so that the vowels are never together = 8! - 6! * 3! = 40,320 - 4,320 = 36,000 ways

E. There are 4 even positions that can be filled by the 3 vowels in 4 * 3 * 2 = 24 ways
Now 5 positions are left, which can be filled in 5! = 120 ways
No. of ways to arrange the letters in a way so that vowels occupy the even positions = 24 * 120 = 2,880 ways

thanks so much anurag, could you please explain D and E a little more?
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by Anurag@Gurome » Tue Sep 20, 2011 2:44 am
fangtray wrote: thanks so much anurag, could you please explain D and E a little more?
D has been solved by using the information given in C. In C, we were asked to find the no. of ways to arrange the letters in a way so that all vowels are together, while in D the vowels should never be together. So, we subtracted the value obtained in C from the the total no. of arrangements without any restriction.

In E, 3 vowels (O, E, U) occupy the even positions. In all there are 8 letters out of which 5 are consonants (C, M, P, T, R). Let us take C for Consonants and V for vowels.
Then,
Odd..Even..Odd..Even..Odd..Even..Odd..Even
C.....V....C....V.....C.....V....C....V

It can be seen that there are 4 positions for placing 3 vowels, so vowels can be placed at even positions in 4! ways.
There are 3 vowels so only 3 positions will be occupied out of 8, so the remaining 5 positions are to be filled with Consonants, which can be done in 5! ways.
Hence, total no. of ways = 4! * 5! = 2,880 ways
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