There are 10 letters and 10 corresponding envelopes with 10 different addresses. Each letter is placed randomly into an envelope. What is the probability that exactly 9 letters are placed in their correct envelopes?

Introduction / Context:
This is a classic problem from the topic of permutations and derangements, which studies how many ways we can place objects so that none or some of them sit in their natural position. Here we have 10 letters and 10 matching envelopes, and we want the probability that exactly 9 letters end up in the correct envelopes. This condition turns out to be impossible, and the explanation reveals why.

Given Data / Assumptions:

• There are 10 distinct letters and 10 distinct envelopes.
• Each envelope has one correct letter that should be placed inside it.
• Letters are placed randomly, one in each envelope, so each permutation of the 10 letters is equally likely.
• We are interested in arrangements where exactly 9 letters are in the correct envelopes.

Concept / Approach:
If exactly 9 letters are in their correct envelopes, that means 9 positions match perfectly with their intended letters. The remaining one letter must then go into the remaining one envelope. However, that last envelope is precisely the correct envelope for that remaining letter, because all other envelopes are already correctly filled. This leads to a contradiction with the requirement that exactly 9 letters be correct and the remaining one be wrong.

Step-by-Step Solution:
Consider 10 letters labelled L1 to L10 and envelopes E1 to E10, where Li is supposed to go into Ei. If exactly 9 letters are correct, then there are 9 indices i where Li is in Ei. That means 9 envelopes are already filled with their correct letters, leaving exactly one empty envelope, say Ej. The only letter not yet placed is Lj, because all other letters L1 to L10 except Lj have already been used in their correct positions. The last remaining letter Lj must go into the only remaining envelope Ej. But Ej is the correct envelope for Lj, so Lj will also be correctly placed. This leads to all 10 letters being correct, not exactly 9, which contradicts the condition. Therefore there is no possible arrangement with exactly 9 correct letters. Number of favourable arrangements = 0. Probability = favourable arrangements / total arrangements = 0 / 10! = 0.

Verification / Alternative check:
We can verify this logic with a smaller example. If there are 2 letters and 2 envelopes and we want exactly one letter to be in the correct envelope, it is not possible. Either both letters are in their correct envelopes, or both are swapped, giving zero cases with exactly one correct placement. This simple example mirrors the same principle that prevents exactly 9 correct matches out of 10.

Why Other Options Are Wrong:
1/10 or 1/9 suggest there is at least one valid arrangement with exactly 9 correct placements, which is not true. 1 would mean every arrangement has exactly 9 correct letters, which is clearly impossible.

Common Pitfalls:
Many learners initially think it is possible to have exactly one letter in the wrong envelope and imagine that last letter as being misplaced. The key is to remember that when all but one envelope are already correctly filled, the final letter has no choice but to go into its matching envelope. Recognizing this logical constraint is essential in derangement based probability questions.

Final Answer:
The probability that exactly 9 letters are placed into their correct envelopes is 0.