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In how many ways can the letters in MATTERS be arranged if no two iden

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Bunuel
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In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   08 Nov 2021, 02:30
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In how many ways can the letters in MATTERS be arranged if no two identical letters can be adjacent?

(A) 1680
(B) 1800
(C) 2514
(D) 2520
(E) 4320
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   08 Nov 2021, 02:59
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Bunuel wrote:
In how many ways can the letters in MATTERS be arranged if no two identical letters can be adjacent?

(A) 1680
(B) 1800
(C) 2514
(D) 2520
(E) 4320


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MATTERS can be arranged in 7!/2! ways which is 2520

Let TT=x

MAXERS can be arranged in 6! ways which is 720

Thus, total ways in which 2 identical letters are not adjacent= Total number of ways-Number of ways in which identical are together=2520-720=1800

Answer (B)
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   08 Nov 2021, 19:31
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MATTERS be arranged if no two identical letters can be adjacent

Total ways in which MATTERS can be arranged = 7!/2! = 2520

Total ways in which MATTERS can be arranged where 2T's are always together = 6! * 2!/2! = 6! = 720

Total ways in which no two T's are together = 2520 - 720 = 1800

IMO B
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   10 Nov 2021, 06:58
I understand why B is the answer, but can someone point out the flaw in the following logic:

MATTERS (all possibilities) = 7! = 5040

MAXERS (where X = TT, all possibilities) = 6! x 2 = 1440

5040 - 1440 = 3600

What did I miss?
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   10 Nov 2021, 07:12
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pchristenson wrote:
I understand why B is the answer, but can someone point out the flaw in the following logic:

MATTERS (all possibilities) = 7! = 5040

MAXERS (where X = TT, all possibilities) = 6! x 2 = 1440

5040 - 1440 = 3600

What did I miss?


Your mistake is highlighted above.
We can arrange n unique objects in n! ways.
So, we can arrange the 6 letters in MAXERS in 6! ways. (not 6! x 2 ways)
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   17 Nov 2021, 09:55
BrentGMATPrepNow wrote:
pchristenson wrote:
I understand why B is the answer, but can someone point out the flaw in the following logic:

MATTERS (all possibilities) = 7! = 5040

MAXERS (where X = TT, all possibilities) = 6! x 2 = 1440

5040 - 1440 = 3600

What did I miss?


Your mistake is highlighted above.
We can arrange n unique objects in n! ways.
So, we can arrange the 6 letters in MAXERS in 6! ways. (not 6! x 2 ways)


In addition to this, MATTERS can be arranged in 7!/2! ways (because "T" occurs twice) and not in 7! ways.

Hope this helps.
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   04 Mar 2022, 13:29
The letters of the word MATTERS can be rearranged in a total of 7!/2! ways which is equivalent to 5040/2 = 2520

The total number of ways in which TT are not together = 2520 - (total number of ways in which TT are together)

Total number of ways in which TT are together can be calculated by consider TT as one unit (K) and rearranging K with M, A, E, R, S. This gives us a total of 6! or 720 different arrangements.

Hence, required answer = 2520 - 720 = 1800

Hence, option B.

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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   26 Apr 2023, 07:52
Asked: In how many ways can the letters in MATTERS be arranged if no two identical letters can be adjacent?

M-1
A-1
T-2
E-1
R-1
S-1

Number of ways the letters in MATTERS be arranged if no two identical letters can be adjacent = 7!/2! - 6! = 1800

IMO B
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   26 Apr 2023, 08:00
Bunuel wrote:
In how many ways can the letters in MATTERS be arranged if no two identical letters can be adjacent?

(A) 1680
(B) 1800
(C) 2514
(D) 2520
(E) 4320
\(\frac{7!}{2!} - 6! = 1800\), Answer must be (B)
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink] New post   26 Apr 2023, 19:41
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Bunuel wrote:
In how many ways can the letters in MATTERS be arranged if no two identical letters can be adjacent?

(A) 1680
(B) 1800
(C) 2514
(D) 2520
(E) 4320



To determine the number of ways that MATTERS can be arranged if no two identical letters can be adjacent, we can subtract the number of ways MATTERS can be arranged the two Ts are next to each other from the total number of ways to arrange the word MATTERS.

The total number of ways to arrange the word matters is 7!/2! = 7 x 6 x 5 x 4 x 3 = 42 x 60 = 2,520

The number of ways MATTERS can be arranged the two Ts are next to each other is 6! = 720.

Thus, the number of ways that MATTERS can be arranged if no two identical letters can be adjacent is 2,520 - 720 = 1,800.

Answer: B
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Re: In how many ways can the letters in MATTERS be arranged if no two iden [#permalink]
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