Let $$P_1, P_2, \ldots, P_{15}$$ be 15 points on a circle. The number of distinct triangles formed by points $$P_i, P_j, P_k$$ such that $$i + j + k \neq 15$$, is:

We have 15 distinct points $$P_1 , P_2 , \ldots , P_{15}$$ lying on a circle. Because no three points on a circle are collinear, every choice of three different points gives a triangle.

The total number of ways to choose any three points from 15 is obtained from the combination formula $$^{n}C_{r} = \dfrac{n!}{r!\,(n-r)!}.$$ Substituting $$n = 15$$ and $$r = 3$$ gives

$$$^{15}C_{3} = \dfrac{15!}{3!\,12!} = \dfrac{15 \times 14 \times 13}{3 \times 2 \times 1} = 455.$$$

So, without any restriction, $$455$$ triangles can be drawn.

Now we must remove those triangles whose indices satisfy the extra condition $$i + j + k = 15$$ (because such triples are not allowed; the question says $$i + j + k \neq 15$$).

To avoid double-counting we consider unordered triples with $$i < j < k$$. We list all sets of three distinct positive integers between 1 and 15 whose sum is exactly 15.

Start with the smallest possible first index.

When $$i = 1$$ we need $$j + k = 14$$ with $$1 < j < k$$.
Possible pairs are $$\{2,12\}, \{3,11\}, \{4,10\}, \{5,9\}, \{6,8\}.$$
That gives 5 triples: $$$(1,2,12),\,(1,3,11),\,(1,4,10),\,(1,5,9),\,(1,6,8).$$$

When $$i = 2$$ we need $$j + k = 13$$ with $$2 < j < k.$$ Pairs are $$\{3,10\}, \{4,9\}, \{5,8\}, \{6,7\},$$ giving 4 triples: $$$(2,3,10),\,(2,4,9),\,(2,5,8),\,(2,6,7).$$$

When $$i = 3$$ we need $$j + k = 12$$ with $$3 < j < k.$$ Pairs are $$\{4,8\}, \{5,7\},$$ giving 2 triples: $$(3,4,8),\,(3,5,7).$$

When $$i = 4$$ we need $$j + k = 11$$ with $$4 < j < k.$$ The only possible pair is $$\{5,6\},$$ giving 1 triple: $$(4,5,6).$$

When $$i \ge 5$$ the smallest possible sum is $$5 + 6 + 7 = 18 > 15,$$ so no further solutions occur.

Thus the total number of disallowed triples is $$5 + 4 + 2 + 1 = 12.$$

Finally, subtract these 12 forbidden triangles from the 455 total triangles:

$$455 - 12 = 443.$$

Hence, the correct answer is Option D.