**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

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