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[[2. What is the coefficient of x^3*y^6*z^5 in (x+y+z)^14? Explain in words why your answer is correct.]] | [[2. What is the coefficient of x^3*y^6*z^5 in (x+y+z)^14? Explain in words why your answer is correct.]] | ||
| − | [[3. How many words of length 7 contain both | + | [[3. How many words of length 7 contain both "a" and "b"]] |
[[4. In how many ways can 6 men and 8 women be lined up such that men are not adjacent?]] | [[4. In how many ways can 6 men and 8 women be lined up such that men are not adjacent?]] | ||
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[[16. Find a recurrence for making a row of colored tiles, colors being red, green, gray. What if red tiles cannot be adjacent? What are the initial conditions? (Note: how many strings are there of length zero?)]] | [[16. Find a recurrence for making a row of colored tiles, colors being red, green, gray. What if red tiles cannot be adjacent? What are the initial conditions? (Note: how many strings are there of length zero?)]] | ||
| − | [[17. How many permutations of the English alphabet do contain | + | [[17. How many permutations of the English alphabet do contain "fish" but not "rat"?]] |
[[18. Prove by induction that 3*11^n + 2*6^n is divisible by 5.]] | [[18. Prove by induction that 3*11^n + 2*6^n is divisible by 5.]] | ||
| − | [[19. Find a recurrence for the number of strings using the letters a,b,c,d that do not have | + | [[19. Find a recurrence for the number of strings using the letters a,b,c,d that do not have "cd" nor "dd" in them. (Hint: start at the end.)]] |
[[20. Solve a_n = 4*a_(n-1) -4*a_(n-2) with a_0=3, a_1=4.]] | [[20. Solve a_n = 4*a_(n-1) -4*a_(n-2) with a_0=3, a_1=4.]] | ||
| − | [[21. Let f_i be the n-th Fibonacci number: f_0=0, f_1=1, f_2=1,... Prove that f_1 + f_3 + f_5 + ... + f_ | + | [[21. Let f_i be the n-th Fibonacci number: f_0=0, f_1=1, f_2=1,... Prove that f_1 + f_3 + f_5 + ... + f_(2n+1) = f_(2n+2).]] |
[[22. Find the generating function for the sequence a_n where a_n is the sum of the squares 1^2+...+ n^2.]] | [[22. Find the generating function for the sequence a_n where a_n is the sum of the squares 1^2+...+ n^2.]] | ||
Revision as of 15:54, 9 March 2009
Sample questions for the midterm:
1. How many bitstrings of length 10 have exactly four zeros?
3. How many words of length 7 contain both "a" and "b"
4. In how many ways can 6 men and 8 women be lined up such that men are not adjacent?
5. How many strings of 5 digits without repetitions contain 1 or 2 but not both?
8. How many words of length 6 are there when adjacent letters being equal is not allowed?
9. How many solutions are there to x+y+z+w = 30 if x is between 5 and 10 and y is at least 6?
10. Find the probability of getting 3 of a kind but nothing better.
11. What is the probability that 2 people play poker against each other and both get 4 of a kind?
15. Find the recurrence for bitstrings that contain 0.
17. How many permutations of the English alphabet do contain "fish" but not "rat"?
18. Prove by induction that 3*11^n + 2*6^n is divisible by 5.
20. Solve a_n = 4*a_(n-1) -4*a_(n-2) with a_0=3, a_1=4.
23. Find the generating function for the Fibonacci sequence.
27. How many numbers between 1 and 10000 are not divisible by any of 5, 7, 11?
