How many rearrangements of abcd are there in which no two adjacent let

Question

How many rearrangements of abcd are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either ab or ba.

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

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zhanbo · Answer

Well, a glance over the answer choices indicates there are not many possibilities.

At first, I thought that I have found 4 arrangements:
ACBD DBCA  Wrong answers 
BDAC CADB

But it turns out there are only two arrangements: 
BDAC CADB

So, the answer should be  (C) . 
(I chose (E) at first.)

av1901 · Answer

abcd has 4! or 24 possible arrangements. 6 arrangements each starting with a,b,c,d

1. Starting with a:

ab with cd and da  {ab together} 
ac with bd and db  {cb/cd together} 
ad with bc and cb  {bc together}

NONE from this set starting from A

2. Starting with b:

ba with cd and dc  {ba together} 
bc with da and ad  {bc together} 
bd with ac and ca  {bdca NO, bdac YES}

1 from this set starting from B

3. Starting with c:

ca with bd and db  {cabd NO, cadb YES} 
cb with ad and da  {cb together} 
cd with ab and ba  {cd together}

1 from this set starting from C

4. Starting with d:

da with bc and cb  {bc together} 
db with ca and ac  {bc/ba together} 
dc with ab and ba  {dc/ab/ba together}

NONE from this set starting from D

Total = 1+1 = 2

Answer - C

ThatDudeKnows · Answer

XXXX

We can't put B in either of the middle positions, as there would only be the far end at which to put both A and C and we can't put two things in one slot. That is, XBX X  would require A and C to both be in the red X in order to avoid putting them next to B...can't do that, so B can't go in the middle. 
Similarly, we can't put C in either of the middle positions, as there would only be the far end at which to put both B and D and we can't put two things in one slot.
So, B and C must go on the ends.

BXXC can only be filled in as BDAC.
CXXB can only be filled in as CADB.

2.

Answer choice C.

bumpbot · Answer

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How many rearrangements of abcd are there in which no two adjacent let

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How many rearrangements of abcd are there in which no two adjacent let

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