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Six cards and six envelopes are labeled with the numbers 1 through 6. Each card must be placed into a different envelope such that no card goes into the envelope with the same number. Additionally, the card labeled 1 must always go into the envelope labeled 2. How many such arrangements are possible?
- 264
- 265
- 53
- 67
Correct answer: 53
Solution
The card labeled 1 is fixed in envelope 2, leaving 5 cards and envelopes for derangement. The number of derangements of 5 items is 44, and multiplying by the fixed arrangement gives 53 valid configurations.
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