Suppose men and women are to be seated in a row so that no two women sit together. if , show that the number of ways in which they can be seated is:

Firstly, we'll fix the postions where the women will be. We have forms to do that. So, we'll obtain a row like:

The n+1 spaces represented by the underline positions will receive the men of the row. Then,

Since there is no women sitting together, we must write that . It guarantees that there is at least one man between two consecutive women. We'll do some substitutions:

The equation (i) can be rewritten as:

We obtained a linear problem of non-negative integer solutions in (ii). The number of solutions to this type of problem are known:

[I can write the proof if you want]

Now, we just have to calculate the number of forms to permute the men that are dispposed in the row:

Multiplying all results:

y=mx+b

step-by-step explanation:

m slope, b y-intercept