After the typist writes 12 letters and addresses 12 envelopes, she inserts the…

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After the typist writes 12 letters and addresses 12 envelopes, she inserts the letters randomly into the envelopes (1 letter per envelope). What is the probability that exactly 1 letter is inserted in an improper envelope? 

  1. A.

    1

  2. B.

    0

  3. C.

    0.5

  4. D.

    0.25

Attempted by 22 students.

Show answer & explanation

Correct answer: B

Step-by-Step Solution

The problem asks for the probability that exactly one letter is inserted into an improper (incorrect) envelope when 12 letters are randomly placed into 12 corresponding addressed envelopes.

  1. Analyze the logical constraint:

    • Suppose we have 12 letters, L1, L2, ..., L12, and 12 corresponding envelopes, E1, E2, ..., E12.

    • If we place 11 letters into their correct envelopes (e.g., L1 into E1, L2 into E2, ..., L11 into E11), there is only one letter (L12) and one envelope (E12) remaining.

    • The last letter (L12) must go into the last envelope (E12).

    • Therefore, if 11 letters are in the correct envelopes, the 12th letter must also be in the correct envelope.

  2. Conclusion:

    • It is logically impossible for exactly 11 letters to be correct and 1 letter to be incorrect. If 11 are correct, the 12th is forced to be correct as well.

    • Because you cannot have exactly one improper insertion, the number of such arrangements is 0.

    • Probability = (Number of favorable outcomes) / (Total number of outcomes) = 0 / 12! = 0.

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