Arrangements in a line
An arrangement records positions. Swapping two different objects makes a new arrangement, but swapping identical objects does not.
Identical objects hide swaps
Pretend A₁ and A₂ are different
A₁A₂B · A₂A₁B · A₁BA₂
A₂BA₁ · BA₁A₂ · BA₂A₁
remove the invisible A swaps ↓
visible
AAB
visible
ABA
visible
BAA
3! ÷ 2! = 3 arrangements
Arrange distinct objects in a line
All distinct objects in a line can be arranged in ways.
Worked example
Four different letters
For A, B, C and D, the four positions have 4, 3, 2 and 1 choices.
This rule is for a line. Circular arrangements are not part of the 9709 Probability & Statistics 1 syllabus.
Divide out invisible swaps
The number of distinct arrangements is
- Begin with as if every object had a label.
- If objects are identical, each visible arrangement was counted times. Divide by .
- Divide separately for every repeated group.
Worked example
Two real-paper word arrangements
ALGEBRAIC has 9 letters with A repeated twice. (9709/52/O/N/24 Q2(a))
REGENERATE has 10 letters with E repeated four times and R repeated twice. (9709/52/M/J/24 Q7(a))
Common mistake
Treat labelled seats as distinct positions
Questions may place people in two or more rows. A front-row seat and a back-row seat are different positions.
Worked example
A pair together in the front row
Ten club members stand in a back row of 6 and a front row of 4. Olly and Petra must stand next to each other in the front row. (9709/52/F/M/24 Q6(c))
- The pair can start in positions 1, 2 or 3 of the front row: 3 choices.
- Olly and Petra can swap: 2 choices.
- Arrange the other 8 people in the 8 remaining seats.
Examiner note
Find the number of distinct arrangements of the letters in STATISTICS.
Show worked answer
There are 10 letters: S appears 3 times, T appears 3 times and I appears twice.