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

+4 votes

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?

posted Sep 22, 2015 by anonymous

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4 Answers

+2 votes

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 Sep 23, 2015 by Maninder Bath
+1 vote
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

answer Sep 22, 2015 by Salil Agrawal
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. :)
+1 vote

IN 5040 WAYS.

answer Oct 2, 2015 by Naga Jyothi
+1 vote

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

answer Dec 30, 2016 by Kewal Panesar

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