Use the notation 𝑃𝑛 𝑟 to represent the number of ways of selecting 𝑟 objects from 𝑛 distinct objects where order is important. 𝑃𝑟 𝑛 = 𝑛! (𝑛−𝑟)! = 𝑛 × (𝑛 − 1) × (𝑛 − 2) × … × (𝑛 − 𝑟 + 1)

A security system requires a 4-digit access code where each digit must be different. The diagram below shows the keypad layout with 8 available digits. Calculate the total number of possible 4-digit codes that can be formed.

A security system requires users to enter a 5-digit access code using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, where each digit can be used at most once. (a) Use permutation notation to express the total number of possible 5-digit codes that can be formed. (1 mark) (b) Determine the numerical value of the total number of possible codes. (1 mark) (c) If the first digit of the code must be even and non-zero, determine how many valid access codes can be formed. (1 mark)

A sports club has 9 members who are eligible for executive positions. a) Use permutation notation to write an expression for the number of ways a president, vice-president, and secretary can be selected from these members. (1 mark) b) Determine the value of your expression from part a). Show evidence of the values substituted. (1 mark) c) If the president must be selected from 4 senior members only, while the vice-president and secretary can be selected from any of the remaining 8 members, determine the number of ways these three positions can now be filled. (1 mark)

A photographer is arranging 8 distinct award trophies in a display case that has 5 positions in a single row. Calculate the number of different arrangements possible using the permutation formula P_n^r.

A school debating team has 10 distinct members. The team must select 4 members to fill the positions of First Speaker, Second Speaker, Third Speaker, and Reply Speaker for an upcoming debate. Calculate the total number of different ways these four positions can be filled if any member can be assigned to any position.

Use the formula $P_n^r = \frac{n!}{(n-r)!}$ to determine the number of different ways that 4 students can be selected from a group of 9 students to fill the positions of President, Vice-President, Secretary, and Treasurer of a mathematics club, where each student can hold only one position.

A regional athletics carnival requires officials to assign lane positions for the 100 m sprint final. Eight athletes have qualified, but only six lanes are available at the venue. (a) Use permutation notation to express the number of different ways the six lane positions can be assigned to the eight athletes. (1 mark) (b) Determine the total number of different lane assignments possible. (1 mark) (c) Two of the athletes, Chen and Priya, are from the same training squad. Determine how many of the possible lane assignments have Chen and Priya occupying adjacent lanes. (1 mark)

A school council consists of 12 distinct members. Leadership positions must be filled for President, Vice-President, Secretary, and Treasurer. All 12 members are willing to accept either President or Vice-President. Exactly 8 specific members (members 1–8) are willing to accept Secretary, and exactly 9 specific members (members 1–9) are willing to accept Treasurer. Each member can hold at most one position. Calculate the total number of different ways the four leadership positions can be filled.