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
