Number of solutions of inequality

Question

I have the following combinatorics problem which I have no clue how to solve.

What is the number of solutions of the inequality $$\sum\limits_{i = 1}^4 {{x_i} \ge 0} , {x_i} \in \{ - 1,1\} $$ I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem but this one leaves me pretty much clueless. I thought of somehow transforming the problem to an equivalent problem: $$\sum\limits_{i = 1}^4 {{k_i} \ge 2} ,{k_i} \in \{ 0,2\} $$ but I'm still stuck...Any help will be much appreciated :)

Answer 1

HINT: All four variables or three of them or at least two of them must be equal to $1$.

Answer 2

I know how to solve these kind of questions when the sum is equal rather than greater or equal than, and it equals to a number other than 0, which is a simple stars and bars problem

A lot of mathematics is about recognising how to transform something you don't know how to solve into something you do know how to solve. If you know how to solve $$\sum_{i=1}^4 x_i = k, x_i \in \{-1,1\}$$ then can you substitute the solution into $$\sum_{k \ge 0} \sum_{i=1}^4 x_i = k, x_i \in \{-1,1\}$$ to solve the original problem?

(Note: $k=0$ isn't really a special case after all).