Question

1- A bank requires customers to create a five-digit PIN in order to use a debit card at the bank’s ATMs. How many different PINs can a customer create provided: a. (3 points) there are no restrictions on the digits used? b. (3 points) no digit may be repeated in the PIN? c. (3 points) adjacent digits must be different?

2- A student organization has three officers and 15 members who are not officers. a. (3 points) How many ways can a committee of three people be chosen from all members? b. (4 points) How many ways can a committee of four people be chosen with two members being officers and the other two members being non-officers? c. (4 points) Elections for officers for the following semester are about to occur. None of the current officers are allowed to run for office for the next semester. If all the current non-officers are eligible, how many different ways can a president, vice-president, and treasurer be chosen?

Answer 1

Question 1 a) No restrictions on the digit used 10 10 10 10 10 There are 5 digits required for a PIN. Each of the digits can be filled by any 10 digits available (0,1,2,3,4,5,6,7,8,9) as repetition is allowed Hence the total number of ways = 10*10*10*10*10 = 105 b) No repetition 7 10 9 00 7 There are 5 digits required for a PIN. The first place can be filled in 10 ways and subsequently each of the places can be filled by one less digit Hence the total number of ways = 10*9*8*7*6 = 30,240 c) Adjacent digits must be different 10 9 9 9 9 There are 5 digits required for a PIN. The first place can be filled in 10 ways and its adjacent can be filled in 9 ways as the adjacent has to different Hence the total number of ways = 10*9*9*9*9 = 65,610

Question 2 Officers 3 Non-Officers 15 a) Committee of 3 people Total members available = 3+15 = 18 We need to select 3 members out of 18, which is given as 18! = 816 3! * 15! b) Four people with two members being officers and the other two members being non-officers Number of ways of selecting 2 officers 3! 2! 1! = 3 15! Number of ways of selecting 2 non-officers = = 105 2!*13! Total number of ways = 105*3 = 315 c) Different ways to select a president, vice-president, and treasurer from non-officers First, we select 3 people out of 15 from non-officers = 15! 3!*12! = 455 Now, these 3 people can be assigned any position of president, vice-president, and treasurer Hence total number of ways = 455*3! = 455*6 = 2,730