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The letters C, I, R, C, L, and E can be used to form 6-letter strings

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The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 21 Sep 2019, 03:16
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The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240


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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 21 Sep 2019, 03:25
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gmatt1476 wrote:
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240


PS54110.01


Good arrangements = total arrangements - bad arrangements.

Total arrangements:
Number of ways to arrange 6 elements = 6!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! to account for the two identical C's:
6!/2! = 360.

Bad arrangements:
In a bad arrangement, the two C's are in adjacent slots.
Let [CC] represent the 2 adjacent C's.
Number of ways to arrange the 5 elements [CC], I, R, L and E = 5! = 120.

Good arrangements:
Total arrangements - bad arrangements = 360-120 = 240.


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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 21 Sep 2019, 05:18
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Total possible arrangements for CIRCLE ; 6!/2! ; 360
and arrangements for CCIRLE ; CC; X ; XIRLE; 5! ; 120
so different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter = 360-120 ; 240
IMO E

gmatt1476 wrote:
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240


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The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 26 Oct 2019, 09:54
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gmatt1476 wrote:
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240



We can use the rule that says: TOTAL number of outcomes if we IGNORE the rule = (number of outcomes that FOLLOW the rule) + (number of outcomes that BREAK the rule)
In other words: Number of ways to arrange the 6 letter if we IGNORE the rule = (number of words that DON'T have adjacent C's) + (number of words that DO have adjacent C's)

Rearrange to get: number of words that DON'T have adjacent C's = (Number of ways to arrange the 6 letter if we IGNORE the rule) - (number of words that DO have adjacent C's)

Number of ways to arrange the 6 letter if we IGNORE the rule
If we IGNORE the rule, then we are arranging the letters in CIRCLE
Since we have DUPLICATE letters, we can apply the MISSISSIPPI rule (see video below)

In the word CIRCLE:
There are 6 letters in total
There are 2 identical C's
So, the total number of possible arrangements = 6!/(2!) = 360

number of words that DO have adjacent C's
Take the two C's and "glue" them together to get the SUPER LETTER "CC"
This ensures that the C's are together
We now must arrange CC, I, R, L, E
We can arrange n different objects in n! ways
So, we can arrange CC, I, R, L, and E in 5! ways (= 120 ways)
So, number of words that DO have adjacent C's = 120

So, number of words that DON'T have adjacent C's = 360 - 120 = 240

Answer: E

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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 26 Oct 2019, 15:33
gmatt1476 wrote:
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240


PS54110.01

Letters in word CIRCLE can be arranged in !6/!2 ways, and for the said condition we have to subtract those cases(bundle those two C's together and count them as 1 unit) where two C's are coming together.
!6/!2-!5
=240
E:)
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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 31 Oct 2019, 08:13
Total number of combinations possible= 6!/2=360
Number of combinations with 2 Cs together = 5!=120
Number of combinations with 2 Cs separated = 360-120= 240

Hence E
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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 23 Nov 2019, 10:25
gmatt1476 wrote:
The letters C, I, R, C, L, and E can be used to form 6-letter strings such as CIRCLE or CCIRLE. Using these letters, how many different 6-letter strings can be formed in which the two occurrences of the letter C are separated by at least one other letter?

A. 96
B. 120
C. 144
D. 180
E. 240


PS54110.01


Alternatively, a different approach could be to actually list down the cases
C_ C _ _ _ The intermediate letters can be arranged in 4! ways. C need not come in the first position and can be moved around 4 spaces to the right(keeping space between the C as one) so 4! * 4
C_ _C _ _ 4! * 3 (C's can be shifted only 3 places to the right)
C_ _ _C _ 4! * 2
C_ _ _ _C 4! * 1


Total=4! * 4 + 4! * 3 + 4! * 2 + 4! * 1= 4!*10 = 240
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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 01 Dec 2019, 20:00
Why is it that for this case, we don't need to multiply the 5! by 2! due to each C being able to be in either spot?
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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings  [#permalink]

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New post 07 Dec 2019, 17:06
Hi All,

We're told the letters C, I, R, C, L, and E are to be used to form 6-letter strings such as CIRCLE or CCIRLE. We're asked for the number of different 6-letter strings that can be formed in which the two occurrences of the letter C are separated by AT LEAST one other letter. There are a couple of different ways to approach this question - and if you don't know an elegant way to approach it, then you can still get the correct answer with a little permutation math and some 'brute force.'

If the first letter is a C, then the second letter CANNOT be an C (that second letter would have to be one of the other 4 non-C letters)...

C 4

From here, any of the four remaining letters can be in the 3rd spot. After placing one, any of the remaining three letters can be in the 4th spot, etc. and the last letter would be in the 6th spot...

C 4 4 3 2 1

This would give us (4)(4)(3)(2)(1) = 96 possible arrangements with a C in the 1st spot.

If a non-C is in the 1st spot and a C is in the 2nd spot, then we have...

4 C _ _ _ _

A non-C would have to be in the 3rd spot (3 options), then any of the remaining three letters could be 4th, etc...

4 C 3 3 2 1

This would give us (4)(3)(3)(2)(1) = 72 possible arrangements

Next, we could have two non-Cs to start off, then Cs in following spots...

4 3 C 2 1 C --> (4)(3)(2)(1) = 24 possible arrangements
4 3 C 2 C 1 --> (4)(3)(2)(1) = 24 possible arrangements
4 3 2 C 1 C --> (4)(3)(2)(1) = 24 possible arrangements

This would give us an additional (3)(24) =72 possible arrangements

There are no other options to account for, so we have 96+72+72 = 240 total arrangements.

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Re: The letters C, I, R, C, L, and E can be used to form 6-letter strings   [#permalink] 07 Dec 2019, 17:06
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