Question

If there are 24 ways to arrange the letters A,B,C,D:
a. How many derangements are there? (23?)
b. How many have exactly 1 letter in the correct place? (9?)
c. 2 letters? (4?)
d. 3 letters? (0?)
e. 4 letters? (1?)
f. No letters? (9?)
g. What is the sum of the answers to b-f? (23?)
h. What have and question number is the answer in question g equal to?

Answer 1

http://www.themathpage.com/aprecalc/permutations-combinations.htm

Answer 2

This is what I have

Answer 3

Do you have to list all 9 derangements of ABCD?

Answer 4

Yep. Duh! I was thinking arrangements the whole time (it's the wording of the questions I think. But still not sure about the answers I am getting.

Answer 5

Let's try that again
BADC BCDA BDAC
CADB CDAB CDBA
DABC DCAB DCBA

Answer 6

Got that.

Answer 7

The subsequent questions are asking how many of the ARRANGEMENTS have exactly______ in the correct place?

Answer 8

Okay - well you got it again LOL

Okay so just the first part of the question has to due with derangements.
The rest refer to arrangements.

Answer 9

Right.

Answer 10

I am showing 7 arrangements with exactly one letter in correct place now.

Answer 11

4 with exactly 2 in correct order

Answer 12

I'm working on showing all 24 arrangements of ABCD
I think that will help

Answer 13

I attached those in a picture earlier. It was just missing ABCD

Answer 14

Oh gee, well here they are
A B C D
A B D C
A C B D
A C D B
A D B C
A D C B

B A C D
B A D C
B C A D
B C D A
B D A C
B D C A

C A B D
C A D B
C B A D
C B D A
C D A B
C D B A

D A B C
D A C B
D B A C
D B C A
D C B A
D C A B

Okay I'll look for how many are in the correct oreder.

Answer 15

Okay, I think I have all the correct orders right now and the sum of the =24.

Answer 16

I count 8 with exactly 1 letter in the correct place
A C B D
A D C B
B C A D
B D C A
C A B D
C B A D
D A C B
D B A C

Answer 17

That's what I have.

Answer 18

6 with exactly 2

Answer 19

None with exactly 3

Answer 20

1 with exactly 4

Answer 21

And then the 9 with none.

Answer 22

Hmmmmmmm looks like the first one
A C B D
is wrong - that has 2 in the correct place

Answer 23

Second one also has 2 letters in correct place
ARRGGHHH !!!!!!!!!!!!!!!!!!!!!

Answer 24

Yeah, I did that the first time also. But I still had 8 with one

Answer 25

ACDB ADBC BCAD BDCA
CADB CBDA DACB DBAC

Answer 26

what is the difference between a derangement and an arrangement?

Answer 27

(im just curious)

Answer 28

A derangement has no object in its correct place.

Answer 29

oh

Answer 30

An arrangement could have one or more in the correct place

Answer 31

so you can count these by

no letters in the right place
1 letter in the right place
2 letters in the right place
3 letters in the right place
4 letters in the right place

Answer 32

arrangements are 1 or more letters in the incorrect place

Answer 33

Right. However, In this case, you won't have any with 3 letters in the correct place. Since there are four letters, if three are in correct places, then all four are in the correct place.

Answer 34

An arrangement can also have all values (letters) in the correct place.

Answer 35

good point :)

Answer 36

Derangements - NO letters in correct place 9
1 letter in correct place 9
2 letters in correct place 5
4 letters in correct place 1

Answer 37

Hmmmmm, on my second and third count, I got 8 and 6. :/

Answer 38

Yes I think 8 and 6 is correct

Answer 39

As far as derangements there is a formula so I know that number is correct:
http://www.1728.org/derange.htm
As far as 1 number and 2 numbers, I'm sure there must be a formula but I'm not familiar with it.

Answer 40

The sum would be 24 of course.

Answer 41

bcurley
Do you need the actual arrangements for 1 letter and 2 letters?
(I have those on a separate spreadsheet and it would be VERY EASY to copy and paste.)

Answer 42

So now where I am stuck is the question that says "What value and question number (letter) in the answer in "g" equal to?"

Answer 43

No, no need to list the different variations. Just asks how many.

Answer 44

Okay - as far as the total (I can't see ALL the questions) I imagine the answer is the sum of the derangements, 1 letter 2 letter, etc would equal the total number of arrangements. 24

Answer 45

Hang on...

Answer 46

This shows all the questions as written on the paper.

Answer 47

Okay it is what I said.
The answers to all those questions about the derangements, 1 letter in correct place, etc would equal ALL possible arragements.

Answer 48

She did give us the hint of (Sum). Took me a minute to realize that.

Answer 49

Okay - then I guess we are all set. :-)
Okay see ya.

Answer 50

Thanks. I appreciate it.

Answer 51

u r welcome :-)