Arranging r from n
Sometimes only part of a group of distinct objects is arranged without reuse. Count the ordered positions and stop when all positions are filled.
Fill the ordered roles, then stop
captain
5 choices
vice
4 choices
speaker
3 choices
⁵P₃ = 5 × 4 × 3 = 60
Fill r ordered positions from n objects
Symbols
- = number available (objects)
- = number arranged (positions)
- = ordered arrangements (ways)
By hand, read as descending factors starting from . Here and are whole numbers with .
Worked example
Three-letter codes without repetition
Make a three-letter code from 7 different available letters.
Answer
Common mistake
The expression has exactly descending factors. For , stop after .
Handle a compulsory object before the open slots
Worked example
The oldest student must sit on the sofa
Four of 18 students sit in four labelled seats, and the oldest must sit.
- Choose the oldest student's seat: 4 choices.
- Fill the other 3 seats from the remaining 17 students.
Answer
Key idea
A compulsory object does not mean “multiply by 1” and forget its position. If its position is not fixed, count where it can go.
Use nPr on your calculator
- On most calculators, enter n, choose nPr, then enter r.
- This is the fastest way to evaluate a large permutation after you have decided that order matters and reuse is forbidden.
- Write or the descending product so the method is visible even if a calculator gives the number.
Your turnIndependent check [2 marks]
How many ordered three-letter codes can be made from 8 different letters without repetition?
Show worked answer
Answer