Homework 7 Due 04.03.17 (Monday) It’s all about counting! When answering countin
ID: 3808387 • Letter: H
Question
Homework 7
Due 04.03.17 (Monday)
It’s all about counting! When answering counting problems, be explicit as to how you arrived at your solutions – i.e., write down the arithmetic expression you used to get your answer; if necessary, use words. This way, even if your final answer is wrong, your solution will help us assess whether you’re taking the right approach or not. If yes, you’ll get more partial points.
1. Passwords. Passwords are used by many computer systems to authenticate iden- tity. Users are always advised to choose passwords that are easy to remember but hard to guess. But how?
Make it long and mix it up. That is, use a password that is long enough and makes full use of the characters of a keyboard so that it is computationally infeasible for someone to try all possibilities to get into the system.
a1. There are 94 different characters in a typical keyboard: 52 case- sensitive letters, 10 numbers and 32 other characters. How many different passwords can be formed if a password has to be 8 characters long?
a2. What if a password can be 8 to 12 characters long?
a3. Suppose a system insists that a password cannot consist of just letters like SesameStreet nor just numbers like 123456789. If a password has to be 8 characters long, how many different passwords are acceptable to the system?
a4. What if it has to be between 8 to 12 characters long?
Salt it. Another strategy is to start with a word or phrase that is easy to remember and then add some random changes to it. Bob starts with Superman and then applies one modification to it where a modification consists of inserting a number at some position. This process can produce a string like Super5man.
b1. How many different strings can Bob create starting with Superman?
Suppose Bob wants to apply two modifications to produce a string like 9Super5man or a string like Super95man. We want to determine the dif- ferent strings Bob can create. He does this in two steps:
Step 1. Choose the first number (say 5) and insert it in Superman. (It produces Super5man, etc.)
Step 2. Choose the second number (say 9) and insert it in the string obtained after Step 1. (If the string from step 1 was Super5man, then the result can be 9Super5man or Super95man, etc.)
b2. To determine the number of different strings Bob can create, we can just apply the Product Rule (Handout 8, page 3). Explain why the product
1
rule will result in overcounting. In particular, what strings will be counted more than once?
b3. How would you solve the problem? Explain why you think it is correct.
Sec. 6.3: 4, 6 (b, c, d only), 20.
Team Panther is participating in a basketball game. The team has 12 active players but only five of them play during the game.
a. How many different groups of 5 players can play in the game?
b. Alex and Jordan are the two point guards of the team. How many different groups of 5 players can the coach create if one point guard has to be part of the group?
c. Steph is their star player. How many different groups of 5 players can the coach create if Steph is part of the group and only one of Alex and Jordan can be in the team?
Explanation / Answer
Answering only 1st 4 questions as multiple questions are posted:
a1. Number of possible passwords = 94^8 = 6095689385410816
a2. for 8-12 characters long, number of passwords = 94^8 + 94^9 + 94^10 +94^11 + 94^12 = 481037737488646537669376
a3. Number of possible passwords with only digits = 10^8 = 100000000
Number of passwords using alphabets = 53459728531456
Total number of passwords excluding the above = 6042229556879360
a4. Number of possible passwords with only digits = 10^8+10^9+10^10+10^11+10^12 = 111110000000
Number of passwords using alphabets>
398541260467162000000
Number of possible passwords = 481037737488646537669376 - 111110000000 -
398541260467162000000 =
398541260467162000000
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