# tut01 - The University of Sydney MATH 1004 Second Semester...

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The University of Sydney MATH 1004 Second Semester Discrete Mathematics 2011 Tutorial 1 Week 2 1. Which of the following strings of brackets are balanced? In each case, explain carefully why the string is, or is not, balanced: ( i ) ((()() ( ii ) (())() ( iii ) ())(() 2. Suppose you have numbers x 0 , x 1 , x 2 , . . . , x n and you want to multiply them together. In how many ways can you insert brackets into the string x 0 x 1 x 2 . . . x n so that the order of multiplication is completely speciFed? Each pair of brackets should contain just two terms. ±or example, when n = 2, there are two ways: ( x 0 ( x 1 x 2 )) and (( x 0 x 1 ) x 2 ). Try n = 3 and n = 4. 3. Consider a river system with n sources which eventually merge to form a single stream. Assuming that no more than two streams merge at any point, we are interested in the number of ways that the mergers can take place. ( i ) Compile a table of values for n = 1, 2, 3, and 4 and then Fnd (guess) a general formula.

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tut01 - The University of Sydney MATH 1004 Second Semester...

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