A secretary types 4 letters and then addresses the 4 corresponding envelopes. In how many ways can the secretary place

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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

[spoiler]OA=B[/spoiler]

Source: Princeton Review

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Vincen wrote:
Wed Jun 24, 2020 2:31 am
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

[spoiler]OA=B[/spoiler]

Source: Princeton Review
When we scan the answer choices (ALWAYS scan the answer choices before choose your plan of attack), we see that all of the answer choices are relatively small.
So, a perfectly valid approach is to list and count the possible outcomes

Let a, b, c and d represent the letters, and let A, B, C and D represent the corresponding addresses.

So, let's list the letters in terms of the order in which they are delivered to addresses A, B, C, and D.
So, for example, the outcome abcd would represent all letters going to their intended addresses.
Likewise, cabd represent letter d going to its intended address, but the other letters not going to their intended addresses.

Now let's list all possible outcomes where ZERO letters go to their intended addresses:
- badc
- bcda
- bdac
- cadb
- cdab
- cdba
- dabc
- dcab
- dcba

DONE!
There are 9 such outcomes.

Answer: B
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Vincen wrote:
Wed Jun 24, 2020 2:31 am
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

[spoiler]OA=B[/spoiler]

Solution:

Let’s call the 4 letters A, B, C and D. Assume the correct order is ABCD. That is, letter A goes to the first envelope, B to the second, C to the third and D to the fourth.

Since A can’t be placed into the first envelope (otherwise, it’s in the correct envelope), we can list all the ways where A is not in the first envelope:

BACD, BADC, BCAD, BCDA, BDAC, BDCA

CABD, CADB, CBAD, CBDA, CDAB, CDBA

DABC, DACB, DBAC, DBCA, DCAB, DCBA

Now, let’s also eliminate those that have at least one letter in the correct envelope (for example, if B is in the second position, it’s in the correct envelope):

BACD, BADC, BCAD, BCDA, BDAC, BDCA

CABD, CADB, CBAD, CBDA, CDAB, CDBA

DABC, DACB, DBAC, DBCA, DCAB, DCBA

We see that the ones that have at least one letter in the correct envelope are in bold; everything else has no letters in the correct envelopes. Therefore, there are 9 ways the secretary can place the letters in the envelopes so that NO letter is placed in its correct envelope.

Answer: B

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