Puzzle

Hi I was given a puzzle to work out and failed miserably, so I was given the answer but no explanation and it is still bugging me.

I may have also explained it incorrectly.

A man writes 7 letters to 7 people. He addresses 7 different envelopes. How many combinations are there that he would not have all the letters in their correct envelopes?.

I have been told the answer is 1854. Does anyone know how this answer is arrived at?.

Not have ANY of the letters in their correct envelopes. The answer is the number of derangements for 7 objects, D(7) = 1854. (A derangement is any such combination.) D(1) is obviously zero. Other derangements can be found from the recursive expression:

D(n) = n*D(n-1) + (-1)^n

Duncan Smith

Incorrectly, I'd say.

7 letters ('to 7 people' is redundant. The 7 different addresses says the same thing.)

OK, 7 letters and 7 envelopes.

Since each envelope has a different address, they are distinguishable. Line them up, alphabetically if you want.

7 choices for the letter to go in the first one. 6 choices for the second, 5 for the third, etc.

That means 7! arrangements, which is 5040 arrangements. Only one of which is correct, so there are 5039 incorrect arrangements.

QED, sort of.

Hi

Thanks but I was told that the answer was 1854 and the correct answer. Duncan has been the first person to agree with me.

Ron

Yeah, Duncan's pretty smart. Quick, too.

Actually, riverman and Duncan are both right.

Duncan calculated that there are 1954 ways to get *all* of the letters in an incorrect envelopes, riverman calculated there are 5040 ways to get *at least one* letter in the wrong envelope.

In your original post, you asked how many ways to 'not have all the letters in their correct envelopes?' This is asking how to have at least one letter in an incorrect envelope. It's possible that you copied it from some other source, which asked how many ways to 'have all the letters not in their correct envelopes?' This is asking how to have zero letters in the correct envelope, a very different question.

Below. 'n' is in cell A1.

Duncan

n D(n)

1 0 =-1^A2

2 =A3*B2+C3 =-1^A3

3 =A4*B3+C4 =-1^A4

4 =A5*B4+C5 =-1^A5

5 =A6*B5+C6 =-1^A6

6 =A7*B6+C7 =-1^A7

7 =A8*B7+C8 =-1^A8

8 =A9*B8+C9 =-1^A9

9 =A10*B9+C10 =-1^A10

10 =A11*B10+C11 =-1^A11

11 =A12*B11+C12 =-1^A12

12 =A13*B12+C13 =-1^A13

13 =A14*B13+C14 =-1^A14

14 =A15*B14+C15 =-1^A15

15 =A16*B15+C16 =-1^A16

Yes. I realised the question was (strictly speaking) wrong when I saw the proposed answer.

Well that didn't paste too well. Try again.

n D(n) 1 0 =-1^A2 2 =A3*B2+C3 =-1^A3 3 =A4*B3+C4 =-1^A4 4 =A5*B4+C5 =-1^A5 5 =A6*B5+C6 =-1^A6

Klein bottle for rent - enquire within.

Both inquire and enquire are valid spellings, but Merriam Webster gives an actual definition with inquire, while for enquire, it says 'variant of inquire.'