Calculating combinations Of marketing strategies

Question

if I have a list of 10 different marketing strategies and can use any combination of them from 1 to all 10, how many possible combinations are there?

Is it $10\cdot 10$?

Answer 1

It's actually $2^{10} - 1$, which is the number of non-empty subsets of the set $$\{s_1,s_2,s_3,s_4, s_5, s_6, s_7, s_8, s_9, s_{10}\}$$

Every combination of the strategies can be written as a string of ones and zeroes, where you write $1$ in the $i$-th place if you yill use strategy $s_i$. So, if you use only strategies $s_1, s_3$ and $s_7$, you write $$101000100$$

Now, you count how many such strings you can write.

You can write $0$ or $1$ on the first place, so there are $2$ options. And independently of that, you have $2$ options on the second space, so all together, $4$ options. $2$ options on the third place, independent of the first two, make that $2\cdot 2\cdot 2$ options.

Repeating the thinking gets you to $2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 = 2^{10}$ options, but now one of the options included is $0000000000$, so you have one option too many. Therefore, the result is $2^{10}-1$