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MACM 201 Fall 04
Midterm 1
Do not open this booklet until you are told to do so.
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Before the test begins, enter the information above. Read all the questions before starting. You are not allowed to write for the first two minutes of the test. Answer the questions in the order that best suits your strengths, taking into account the number of marks alloted to each question. Full marks will be reserved for answers that are correct in all essential details. Remember that your answers should be in a form that another student could understand without undue effort: a poorly expressed but correct result is not sufficient. This booklet should contain numbered pages 1 to 7, this page being page 1. As soon as the test begins, check to make sure you have all the pages and raise your hand if you do not.
1 2 3 4 5 6 Total
- (a) 1 mark Let A and B be events. Define the conditional probability P (A|B).
(b) 5 marks Each of five friends in turn tosses a coin whose probability of landing heads is p. If we know that not all five tosses land the same, what is the probability that one toss lands differently from all the others?
- Each of the following statements is false. For statements (a), (b) and (c), correct the text contained within the box. For statement (d), demonstrate conclusively that it is false.
(a) 1 mark The number of derangements of n objects is 1 − 1 + (^) 2!^1 − (^) 3!^1 + · · ·
(b) 1 mark The number of integer solutions to the equation x 1 + x 2 + · · · + xn = r, where xi ≥ 0 for each i, is
(n+r− 1 r− 1
(c) 1 mark The Generalised Principle of Inclusion-Exclusion gives an expression for the number of elements of the set satisfying at least m conditions. (d) 3 marks Let S be the set of shaded chessboards comprising 4 squares. Let T be the set of rook polynomials corresponding to the chessboards of S. There is only one polynomial of degree 3 in T.
- (a) 2 marks Define the rook polynomial r(C, x) for a chessboard C consisting of a set of shaded squares, defining any notation you use. (b) 2 marks State a recursive formula for r(C, x), defining any notation you use. (c) We wish to determine the number of 1 − 1 functions f : { 1 , 2 , 3 } → {u, v, w, x} which satisfy the constraints
f (1) 6 = u, x f (2) 6 = v f (3) 6 = u, v, w.
2 marks Set up this problem as an inclusion-exclusion problem by defining a suitable set S and conditions ci, and represent the conditions by means of some chessboard C. 3 marks Use the recursive formula to evaluate the rook polynomial r(C, x) for the chessboard C you have drawn. 2 marks Use the Principle of Inclusion-Exclusion applied to 1−1 functions (which you need not prove or justify), and the calculated value of r(C, x), to determine the number of 1 − 1 functions satisfying the given constraints.
- Note: this question will be marked in accordance with your preference stated below.
2 marks Answer yes or no: would you like to be awarded 2 marks for answering this question?
