Discrete Math

Q1 How many ways are there to pick one number out of 6 choices, choose 2 colors from 7 color options, and permute a sequence of 5 numbers? Q2 Consider the state machine whose states are triples of non-negative integers (r, s, a). The initial state is (x, y, 0). The transitions are given by the rule o that for s > 0: (3r, s/3, a) if 3 | s 8(r, s, a) = (3r, (s - 1)/3, a + r) if 3 | (s - 1) (3r, (s - 2)/3, a + 2r) otherwise. The state machine will always reach a state of the form (r, 1, a) in at most log3 y transitions. True or False? Q3 Consider the state machine whose states are triples of non-negative integers (r, s, a). The initial state is (x, y, 0). The transitions are given by the rule & that for s > 0: (3r, s/3, a) if 3 | s 6(r, s, a) = (3r, (s - 1)/3, a+ r) if3 | (s -1) (3r, (s - 2)/3, a + 2r) otherwise. The predicate a = xy is true for every state of this state machine. True or False ? Q4 Choose all of the following that is a strictly decreasing derived variable on the states of the state machine whose states are triples of non-negative integers (r, s, a). The initial state is (x, y, (). The transitions are given by the rule & that for s > 0: (3r, s/3, a) if 3 | s o(r, s, a) = (3r, (s - 1)/3, a + r) if 3 | (s - 1) (3r, (s - 2)/3, a + 2r) otherwise. r is strictly decreasing. s is strictly decreasing a is strictly decreasing. None of the provided values are strictly decreasing. Q5 Suppose that a set B is defined as B ::= B1 X B2 X B3. Choose all of the following that is true. D IB| = \\B1 . B2 . Bal D IB) = Bil . IB21 . |Bal D [ B| = [B1 X B2 X B31