Problem K/C. 873. (October 2025)

K/C. 873. Rabbit has written letters to six of his friends and relations and filled out six envelopes with their addresses. Later he gave the whole stack to Tigger and asked him to put the letters into the envelopes and post them, since Tigger boasted several times that Tiggers are the best at posting letters. However, this was not the case, since Tigger actually couldn't read at all. So, not surprisingly, Tigger will put the six letters into the six envelopes randomly (at least he will make sure that each envelope will contain a single letter). In how many cases will exactly two letters be placed in their respective envelopes?

(5 pont)

Deadline expired on November 10, 2025.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Jelöljük a borítékokat \(\displaystyle A\), \(\displaystyle B\), \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle E\), \(\displaystyle F\) betűkkel; a megfelelő leveleket pedig \(\displaystyle a\), \(\displaystyle b\), \(\displaystyle c\), \(\displaystyle d\), \(\displaystyle e\), \(\displaystyle f\) betűkkel. A két borítékot, amelybe a jó levél kerül 15-féleképpen lehet kiválasztani. (\(\displaystyle A\)-\(\displaystyle B\), \(\displaystyle A\)-\(\displaystyle C\), \(\displaystyle A\)-\(\displaystyle D\), \(\displaystyle A\)-\(\displaystyle E\), \(\displaystyle A\)-\(\displaystyle F\), \(\displaystyle B\)-\(\displaystyle C\), \(\displaystyle B\)-\(\displaystyle D\), \(\displaystyle B\)-\(\displaystyle E\), \(\displaystyle B\)-\(\displaystyle F\), \(\displaystyle C\)-\(\displaystyle D\), \(\displaystyle C\)-\(\displaystyle E\), \(\displaystyle C\)-\(\displaystyle F\), \(\displaystyle D\)-\(\displaystyle E\), \(\displaystyle D\)-\(\displaystyle F\), \(\displaystyle E\)-\(\displaystyle F\)). Legyen mondjuk ez a két boríték az \(\displaystyle A\) és a \(\displaystyle B\). A \(\displaystyle C\), \(\displaystyle D\), \(\displaystyle E\) és \(\displaystyle F\) borítékokba nem kerülhet a saját levelük, amit kilencféleképpen lehet megvalósítani, csakúgy, mint a többi esetben.

Így összesen \(\displaystyle 15 \cdot 9=135\)-féleképpen fordulhat elő, hogy pontosan két borítékba kerül megfelelő levél.

Statistics:

320 students sent a solution.
5 points:	171 students.
4 points:	42 students.
3 points:	52 students.
2 points:	14 students.
1 point:	10 students.
0 point:	14 students.
Not shown because of missing birth date or parental permission:	15 solutions.

Problems in Mathematics of KöMaL, October 2025