Be Excellent To Each Other

Hello, clever people!

Let's suppose Curio could choose up to six numbers between 1 and 16.

So he could choose 2,5,7,8,9 and 10, or maybe 4,5 and 6.

The numbers have to be chosen sequentially, as above. How many options does Curio have?

I assume the numbers also need to be unique?

If not, no idea.
If they do need to be unique, then no idea.

This is permutations! It's probably easier to chop them down into bits - possible ways of choosing 1 number, ways of choosing 2 numbers etc.

So:

Ways of choosing 1 number = 16!/(16 - 1)! = 16
Ways of choosing 2 numbers = 16!/(16 - 2)! = 240
Ways of choosing 3 numbers = 16!/(16 - 3)! = 3360
Ways of choosing 4 numbers = 16!/(16 - 4)! = 43680
Ways of choosing 5 numbers = 16!/(16 - 5)! = 524160
Ways of choosing 6 numbers = 16!/(16 - 6)! = 5765760

So in total - 6337216 ways of choosing.

One, just one option. Curio is terrified of all numbers except for 3 and 5.

No, wait, I've belmed up there.

Ignore those numbers.

Erm in that case there would be 4 options. No numbers picked, 3 picked, 5 picked, 3&5 picked.

It's combinations, not permutations.

Ways of choosing 1 number = 16!/ ( (16 - 1)! * 1! ) = 16
Ways of choosing 2 numbers = 16!/ ( (16 - 2)! * 2! )= 120
Ways of choosing 3 numbers = 16!/ ( (16 - 3)! * 3! )= 560
Ways of choosing 4 numbers = 16!/ ( (16 - 4)! * 4! )= 1820
Ways of choosing 5 numbers = 16!/ ( (16 - 5)! * 5! )= 4368
Ways of choosing 6 numbers = 16!/ ( (16 - 6)! * 6! )= 8008

A grand total of 14892

Plus one more for not choosing any, if that's allowed

That doesn't take in the sequential nature, but that is just a red herring isn't it, I don't think that makes any difference.

Oh. That's a lot less than I thought - but it makes sense.

Yeah, it's just saying that order doesn't matter - you can't choose (1,2) and then (2,1) and have them be different choices.

Can you choose one number or no numbers sequentially?

This is true. I had to change the 8 on my front door to read 3+5