Why do 22 letters make 231 pairs?
A worked explanation of unordered letter pairs, reversed order, repeated letters and final forms.
Start with three letters
Use Aleph, Bet and Gimel. The pairs are Aleph–Bet, Aleph–Gimel and Bet–Gimel. Reversing a pair does not create a new member of this collection. Three letters therefore give three pairs. A drawing with three labelled points makes each relationship visible as one connecting line.
Count choices, then remove duplication
With 22 different letters, the first position has 22 choices. Once it is chosen, the second has 21. This counts 462 ordered pairs. Every unordered pair occurs twice: Aleph–Bet and Bet–Aleph, for example. Divide by two to obtain 231. We exclude a letter paired with itself and count a reversed pair once.
What changes when order matters?
If a question deliberately distinguishes Aleph–Bet from Bet–Aleph, it is asking about 462 ordered pairs of different letters. If repetition is also allowed, two ordered positions have 22 × 22 = 484 possibilities. These are different counting problems. State the rules before attaching an interpretation to the answer.
Why final forms do not increase the count
The five final forms belong to five existing letters. In this 22-letter model they do not introduce five additional letters. A keyboard character count and an alphabetic identity count answer different questions. Keep the object you are counting consistent from the first line to the last.
The 21 Aleph pairings
Aleph can pair with each of the other 21 letters. Those pairings form one part of the complete set. Start an alphabetical list with 21 pairs beginning with Aleph, then 20 beginning with Bet without repeating Aleph, then 19, continuing down to one. The sum 21 + 20 + … + 1 is 231.
Try it with four letters
List all pairs from Aleph, Bet, Gimel and Dalet. There should be six: AB, AG, AD, BG, BD and GD. Check that every pair has two different letters and no reversed duplicate. Then explain why adding a fifth letter adds four new pairs. This exercise checks the counting rule; it does not generate a symbolic interpretation for each pair.
22 × 21 ÷ 2 = 231
These are Babylon study exercises. Keep the source text and its edition distinct from the adopted map.