Anagrams of MATEMATICA

Statement

How many anagrams does the word MATEMATICA have?

Solution

The word has $10$ letters, but some are repeated: A appears $3$ times, M $2$ times, T $2$ times (E, I, C once each). These are permutations with repetition: one starts from $10!$ and divides by the factorials of the multiplicities of the repeated letters. $\frac{10!}{3!\,2!\,2!} = \frac{3\,628\,800}{6\cdot 2\cdot 2} = \boxed{151\,200}$

Links

Topics: Combinatorics
Concepts: Factorial · Permutations
Methods: Factorial permutations
Skills: Combinatorial calculus
Exercise type: Combinatorial problem