55. How many different letter combinations can be obtained by rearranging the letters in the word a) NONNA; b) MATHEMATICA?
55. a) The word NONNA contains 5 letters. If all of them were different, we could obtain $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=120$ different letter combinations from this word. In reality, in each such combination, the letter N is repeated three times. Permuting these three letters among themselves does not change the combination. Since three letters allow 6 permutations, we will get a total of $120: 6=20$ different letter combinations.

b) By analogy with the previous problem, we find that as a result of all possible permutations of the letters in the word MATEМАТИКА, we will have $10!:(2!2!3!)=151200$ different letter combinations.