Problem 6. Prove that
Problem 7. Compute
Problem 8. Consider all words of length in the Latin alphabet. Define the weight of a word as where is the number of letters not used in this word. Prove that the sum of the weight of all words is .
Problem 9. An complex matrix is called t-normal if where is the transpose of . For each , determine the maximum dimension of a linear space of complex matrices consisting of -normal matrices.
Problem 10. Let be a positive integer and let be a polynomial of degree with integer coefficients. Prove that