222m1-practice

# 222m1-practice - 201001 Math 222 Practice Midterm 1...

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201001 Math 222 Practice Midterm 1 Questions Most of these are from actual old midterms or ﬁnals. The rest come from my imagination. The actual midterm will not be this long. 1. Determine the number of integers between 10000 and 99999 with (a) exactly three 7’s; (b) between three and ﬁve even digits (0, 2, 4, 6, 8); (c) no repeated digits. 2. A combination lock has a dial that rotates and has with positions num- bered 0 , 1 ,..., 60. Opening such lock is a process that consists of three steps: (a) Turn the dial clockwise, passing zero twice and stopping at x 1 ; (b) Turn the dial counterclockwise, passing zero once and stopping at x 2 6 = x 1 ; (c) Turn the dial clockwise, stopping at a number bigger than x 2 before passing zero. A valid combination is a triple of numbers ( x 1 ,x 2 ,x 3 ) for which it is possi- ble to open the lock using the instructions above (for example, ( x 1 ,x 2 ,x 3 ) = (5 , 0 , 58)). Determine the number of valid combinations for this type of lock. 3. How many subsets of

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222m1-practice - 201001 Math 222 Practice Midterm 1...

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