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| − | 1. How many bitstrings of length 10 have exactly four zeros? | + | [[1. How many bitstrings of length 10 have exactly four zeros?]] |
| − | 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 ``a'' and ``b''? | + | [[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?]] |
| − | 5. How many strings of 5 digits without repetitions contain 1 or 2 but not both? | + | [[5. How many strings of 5 digits without repetitions contain 1 or 2 but not both?]] |
| − | 6. In how many ways can one travel from (0,0) to (8,11) going only East or North, and while passing through (4,7) ? | + | [[6. In how many ways can one travel from (0,0) to (8,11) going only East or North, and while passing through (4,7)?]] |
| − | 7. How many strings of length 13, composed of the letters m, n, p, q and no others, have exactly 3 p's and 4 q's? | + | [[7. How many strings of length 13, composed of the letters m, n, p, q and no others, have exactly 3 p's and 4 q's?]] |
| − | 8. How many words of length 6 are there when adjacent letters being equal is not allowed? | + | [[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? | + | [[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. | + | [[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? | + | [[11. What is the probability that 2 people play poker against each other and both get 4 of a kind?]] |
| − | 12. On a die, 4 has probability 2/7, all others have 1/7. On a second die, 3 has probability 2/7 and all others ahve 1/7. Find the chance of rolling a 7 with this pair of loaded dice. | + | [[12. On a die, 4 has probability 2/7, all others have 1/7. On a second die, 3 has probability 2/7 and all others ahve 1/7. Find the chance of rolling a 7 with this pair of loaded dice.]] |
| − | 13. Toss a coin 10 times. Assume that head shows with 55 % in each roll. Find the probability of getting at least 2 heads in the 10 rolls. | + | [[13. Toss a coin 10 times. Assume that head shows with 55 % in each roll. Find the probability of getting at least 2 heads in the 10 rolls.]] |
| − | 14. Imagine a casino has the following game. Roll a fair die 3 times. You get $ 27 if you roll at least two 2's. Otherwise you get nothing. Find the minimum price the casino should ask for playing this game. | + | [[14. Imagine a casino has the following game. Roll a fair die 3 times. You get $ 27 if you roll at least two 2's. Otherwise you get nothing. Find the minimum price the casino should ask for playing this game.]] |
| − | 15. Find the recurrence for bitstrings that contain 0. | + | [[15. Find the recurrence for bitstrings that contain 0.]] |
| − | 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 ``fish'' but not ``rat''? | + | [[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 ``cd'' nor ``dd'' in them. (Hint: start at the end.) | + | [[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_{2n+1) = f_(2n+2). | + | [[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.]] |
| − | 23. Find the generating function for the Fibonacci sequence. | + | [[23. Find the generating function for the Fibonacci sequence.]] |
| − | 24. Page 472, number 27. | + | [[24. Page 472, number 27.]] |
| − | 25. Page 472, number 31. | + | [[25. Page 472, number 31.]] |
| − | 26. Page 472, number 35. | + | [[26. Page 472, number 35.]] |
| − | 27. How many numbers between 1 and 10000 are not divisible by any of 5, 7, 11? | + | [[27. How many numbers between 1 and 10000 are not divisible by any of 5, 7, 11?]] |
Revision as of 15:50, 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.
[[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).]]
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?
