Given, a + b + c = 1, so that a + b = 1 – c, b + c = 1 – a, c + a = 1 – b
Now, AM > GM ⇒ (a + b) > 2\(\sqrt{ab}\) ⇒ (1 – c) > 2\(\sqrt{ab}\) …(i)
Similarly, (b + c) > 2\(\sqrt{bc}\) ⇒ (1 - a) > 2\(\sqrt{ab}\) …(ii)
(c + a) > 2\(\sqrt{ca}\) ⇒ (1 - b) > 2\(\sqrt{ca}\) …(iii)
Multiplying (i), (ii) and (iii), we get
⇒ (1 – a) (1 – b) (1 – c) > 8 \(\sqrt{ab}\) \(\sqrt{bc}\) \(\sqrt{ca}\)
⇒ (1 – a) (1 – b) (1 – c) > 8 abc ⇒ K = 8.