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# A secretary types 4 letters and then addresses the 4 corresponding env

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Manager
Joined: 19 Feb 2018
Posts: 113
A secretary types 4 letters and then addresses the 4 corresponding env  [#permalink]

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01 Dec 2019, 05:56
00:00

Difficulty:

55% (hard)

Question Stats:

29% (01:33) correct 71% (01:36) wrong based on 7 sessions

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A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15
Manager
Joined: 19 Feb 2018
Posts: 113
Re: A secretary types 4 letters and then addresses the 4 corresponding env  [#permalink]

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01 Dec 2019, 05:57
harsh8686 wrote:
A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15

By the wording, if even 1 letter is in the correct envelope, it counts as a fail.

In all, there are 4!=(4*3*2*1) or 24 possible ways to put the four letters into 4 different envelopes. Now we need to subtract the failures.

Failure will occur if exactly 1 letter is in the correct envelope. Since there are 4 envelops, there are 4 different ways to accomplish this. So, the value is 4 chose 1. Likewise, there are 4 choose 2 ways to insert 2 letters into the correct envelopes, 4 choose 3 ways to insert 3 letters into the 4 envelopes, and 4 chose 4 ways to insert all 4 letters into the correct envelopes.

Let n=4 be the number of envelopes in which at least 1 of n letters are to be incorrectly placed and let m be the total number of ways of doing so. Then, m can be expressed as below.

m=24−[(4C1)+(4C2)+(4C3)+(4C4)]

m=24−[4+6+4+1]=24−15=9

So, there are 9 ways in which the 4 letters can be inserted into the 4 different envelops in which at least 1 is incorrect.
Math Expert
Joined: 02 Sep 2009
Posts: 59622
Re: A secretary types 4 letters and then addresses the 4 corresponding env  [#permalink]

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01 Dec 2019, 06:06
harsh8686 wrote:
A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place the letters in the envelopes so that NO letter is placed in its correct envelope?

A) 8
B) 9
C) 10
D) 12
E) 15

Discussed here: https://gmatclub.com/forum/a-secretary- ... fl=similar
_________________
Re: A secretary types 4 letters and then addresses the 4 corresponding env   [#permalink] 01 Dec 2019, 06:06
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