A route is just a list of moves: On a grid you may only move right or up. Every shortest route is a list of R's and U's — so just count how many ways you can arrange those moves.
IB-style question — shortest routes on a grid
A robot starts at the bottom-left corner of a grid that is 4 blocks wide and 3 blocks tall. It moves only right or up to reach the top-right corner.
How many shortest routes are there?
Step by step
- To cross the grid it makes 4 right moves and 3 up moves — 7 moves in total.
- A route is an arrangement of R R R R U U U. Just choose which 3 of the 7 moves are 'up'.
Final answer
35 routes.
Each route is an arrangement of 4 right and 3 up moves: ⁷C₃ = 35.
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Divide away the swaps you can't see: When some items are identical, plain n! over-counts — swapping two identical letters looks the same.
Fix it by dividing by the factorial of each repeat: n! ÷ (repeats!).
IB-style question — arrangements of a word
How many different arrangements are there of the letters of the word BANANA?
Step by step
- BANANA has 6 letters. Count the repeats: A appears 3 times, N appears 2 times, B once.
- Divide 6! by the factorial of each repeat.
Final answer
60 arrangements.