I get there are seven slots, but I don’t know how Greg came up with 7!. Aren’t there are only two choices (north or east), and not seven choices, to pick from?
Having said that, I wouldn’t even know how to solve it using my method, because I thought it was 2^7 and I know this is wrong because once Anjali passes north or east by a certain number, they’re forced to pick one direction for the remainder of the way.
In the end she has to end up moving up 4 times and to the right 3 times, so there are 7 instructions/letters for us to re-arrange → NNNNEEE.
It’s not a “choices” problem, its an arrangement problem. How many ways can we arrange NNNNEEE (or in other words, how many different ways can you go north 4 times and right 3 times).
Not necessarily. For example if you go north 3 times already, you cant go north again so we don’t really have 7 choices each with 2 options.
Okay, I think I get it now. I think you saying it’s an “arrangement problem” triggered some thoughts.
For reasons I find hard to explain, arranging “BANANA” and “NNNNEEE” felt like different problems to me, when in fact they’re the same.
Having said that, I still find thinking about this as a “choices” problem more intuitive, because we’re still picking:
An analogy: it’s like covering “NNNNEEE” with distinct coloured boxes (red, blue, yellow, green, purple, orange, white). After arranging them in the seven slots and you uncover the boxes, you realised there are repeats.