In how many ways can all of the letters be placed in the wrong envelopes?

Question

A correspondent writes 7 letters and addresses 7 envelopes, one for each letter. In how many ways can all of the letters be placed in the wrong envelopes?

Answer 1

Answer: 1854

1 letter, 0 way
2 letters, 1 way
3 letters, 2 ways
4 letters: 9 Ways 3*2=6 ways of cycling the 4 around, but then 3 ways of doing 2+2. (=9)

So partitioning is the way to go. A partition of 1 always maps the letter into the right envelope, so there's no answers with partition 1.

7 can be
a "cycle" of 7: with 1 case, which has 6*5*4*3*2 permutations
a "cycle" of 5 plus a "cycle" of 2: with 7*6/2 cases and 4*3*2 permutations
a "cycle" of 4 plus a "cycle" of 3: 7*6*5/6 cases and (2 {for the 3} * 9 {for the 4}) permutations
i.e. 720+504+630=1854

Or we can use formula
7! * ( 1/2! - 1/3! + 1/4! - 1/5! + 1/6! - 1/7! )
= 2520 - 840 + 210 - 42 + 7 - 1
= 1854

Answer 2

E1 E2 E3 E4 E5 E6 E7 
L1 L2 L3 L4 L5 L6 L7

Now assume L1 can be put into 7 ways into an envelope and only one is correct and 6 are wrong ways similarly for L2 there are six ways and so on
So total possible wrong ways are 6*6*6*6*6*6*6
=) 6^7 or 279936

Comments

No. As u said, there are 6 ways in which L1 will not be placed in E1. But for example, say L3 is placed in E1, then in E2 we cannot place both L3 and L2 (Since L3 is already placed in E1), so now the chances reduces to 5 and not.

---

Yes you are right the number of possible ways would be 6! i.e. 6.5.4.3.2.1 :)

Answer 3

7!
IN 5040 WAYS.

Answer 4

Ms Naga Jyothi, please remember that of the 5040 ways only one is true. Hence the answer is 5039