Arranging in a line
- An arrangement in a line means order matters.
- If all objects are different, the count is .
- If some objects look identical, divide by the ways those look-alikes could swap without changing the arrangement.
Line up, then divide the repeats
Distinct objects: . Repeated objects:.
- Start with , as if every object were different.
- For identical copies, divide by .
- Do this for each repeated group. Single objects are , so they do not change anything.
Worked example
First rung: arrange three different letters
- For A, B and C, the first slot has 3 choices.
- After that, the second slot has 2 choices, and the last slot has 1 choice.
Answer
Worked example
One extra step: repeated letters
Arrange the letters in AAB.
- If the two As were different, there would be arrangements.
- The two As can swap without making a new word, so divide by.
Answer
Worked example
Repeated letters in real words
ALGEBRAIC has 9 letters, with A repeated twice. (9709/52/O/N/24 Q2(a))
- Count as if all letters were different, then divide by the A swaps:.
REGENERATE has 10 letters, with E repeated 4 times and R repeated twice. (9709/52/M/J/24 Q7(a))
- Divide by both repeat groups:.
Common mistake
Do not divide by the total number of repeated letters. If a word has A repeated 3 times and N repeated twice, the bottom is, not .