How many ways are there from A to B? Given that you can only move up or move right along the line.
The answer is 16. This one is genuinely hard, and it is hard for a reason worth knowing: the shape is not a neat rectangle. There is an extra square hanging underneath on the left, and that little bump is where people go wrong.
We will build the trick from the smallest possible version and work up.
From the bottom-left corner to the top-right corner of a single square. Only up, only right. How many ways?
Easy — you can see both. Now make it bigger.
Tap through all six routes. Watch how quickly you would start losing track of which ones you had already done.
Six. And on the real exam figure there are 16 — you are not going to draw 16 routes in a 90-minute paper without missing one or drawing one twice.
So here is the trick that means you never draw a single route.
You can only arrive at a dot from below, or from the left.
There is no third way in. Up and right are the only moves.
So the number of ways to reach any dot is just:
ways to get here = ways to the dot below + ways to the dot on the left
Write 1 at A — there is one way to be standing at the start: you have not moved. Then fill in every dot, going along the bottom and up. Try it on the 2-by-2 grid:
Six — the same six you just tapped through, without drawing one single route.
Same rule, same order: bottom row first, then work upwards. Take it one dot at a time and read what each one says.
16.
On the paper that was option A.
Did you notice these two dots?
There is no line underneath them — the bump only exists on the left. So for those dots, "ways from below" is 0, and they just copy the number from their left.
If you forget that and quietly treat the shape as a full rectangle, you get a bigger number. The shape of the figure is part of the question, not decoration.
No score, no timer. Fill in the dots on paper if it helps — that is exactly what you would do in the exam.
A strip 2 squares wide and 1 tall. How many ways from A to B?
A grid 3 wide and 1 tall. How many ways?
A grid 3 wide and 2 tall — a plain rectangle, no bump. How many ways?
A small version of the exam shape — 2 wide, 1 tall, with one square hanging below on the left. How many ways?
1. Put a 1 on A, and a 1 on every dot you can only reach one way — straight along the bottom, straight up the left side.
2. Work bottom row upwards, left to right. Each dot = below + left. If a line is missing on one side, that side gives 0.
3. The number you write on B is the answer — and then read the question again to check it asked for the number of ways.
This is a five-mark method that takes about thirty seconds once your hand knows it. Write the numbers on the figure in the exam paper — that is what the figure is there for.