Anagrams of PRECIPITEVOLISSIMEVOLMENTE

Problem

Compute the number of anagrams of the word PRECIPITEVOLISSIMEVOLMENTE ($26$ letters, with repetitions $P\colon 2$, $R\colon 1$, $E\colon 5$, $C\colon 1$, $I\colon 4$, $T\colon 2$, $V\colon 2$, $O\colon 2$, $L\colon 2$, $S\colon 2$, $M\colon 2$, $N\colon 1$).

Solution

The anagrams of a word with repeated letters are the permutations with repetition: one starts from the factorial of the total number of letters and divides by the factorial of the multiplicities of each letter, so as not to count identical arrangements more than once. With $26$ letters and the given multiplicities: $\boxed{\dfrac{26!}{2!\cdot 1!\cdot 5!\cdot 1!\cdot 4!\cdot 2!\cdot 2!\cdot 2!\cdot 2!\cdot 2!\cdot 2!\cdot 1!}}$ (in the order: $P,R,E,C,I,T,V,O,L,S,M,N$).

Links

Topics: Probabilita
Concepts: Permutazioni con ripetizione
Methods: Anagrammi ripetuti
Skills: Calcolo combinatorio
Exercise type: Problema combinatorio