Explanation:

To solve the problem, let's analyze and simplify:

- We have the ratio \((a + b) : (b + c) : (c + a) = 5 : 7 : 6\).

- Let \(a + b = 5k\), \(b + c = 7k\), and \(c + a = 6k\) for some constant \(k\).

- Solve these equations to express \(a\), \(b\), and \(c\):

- From the equations, \(a = \frac{5k - 7k + 6k}{2} = 2k\).

- \(b = \frac{5k - 6k + 7k}{2} = 3k\).

- \(c = \frac{6k - 5k + 7k}{2} = 4k\).

- Calculate \(a - b + c = 2k - 3k + 4k = 3k\).

- Calculate \(a + b - c = 2k + 3k - 4k = k\).

- The ratio \((a - b + c) : (a + b - c) = 3k : k = 3 : 1\).

Option 3 (3 : 1) is correct.