COMPSCI 1DM3 - Assignment 7
Due date: April 1, 11:59 PM
1. (10 points) Suppose that a “word” is any string of seven letters of the alphabet, with
repeated letters allowed.
(a) How many words begin with A or B or end with A or B?
(b) How many words begin with AAB in some order?
2. (5 points) In a technician’s box there are 400 VLSI chips, 12 of which are faulty. How
many ways are there to pick two chips, so that one is a working chip and the other is
faulty? (Assume that no chips are identical.)
3. (10 points) A club with 20 women and 17 men needs to choose three different members
to be president, vice president, and treasurer.
(a) In how many ways is this possible?
(b) In how many ways is this possible if women will be chosen as president and vice
president and a man as treasurer?
4. (10 points) Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit.
How long will it take to check every possible character combination, if each check takes
one unit of time.
5. (15 points) Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five
elements, the sum of which is even.
6. (15 points)
The figure at the right shows a 4-block by 5-block grid of streets.
Find the number of ways in which you can go from point A to
point B, where at each stage you can only go right or up. (You
are not allowed to go left or down.) For example, one allowable
route from A to B is:
Right, Right, Up, Right, Up, Up, Right, Right, Up.
B
A
7. (10 points) A factory makes automobile parts. Each part has a code consisting of a letter
and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts.
Prove that there are at least three parts that have the same serial number.
8. (10 points) You pick cards one at a time without replacement from an ordinary deck of 52
playing cards. What is the minimum number of cards you must pick in order to guarantee
that you get
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(a) a pair (for example, two kings or two 5s).
(b) three of a kind (for example, three 7s).
9. (15 points) How many permutations of 12345 are there that leave 3 in the third position
but leave no other integer in its own position?
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