1. Total number of triangles from 15 points on a circle

Any 3 distinct points on a circle form a triangle, so the total number is

$\binom{15}{3} = \frac{15\cdot 14\cdot 13}{6} = 455.$

2. Interpret the condition

We need triangles formed by points $P_i,P_j,P_k$ such that

$i+j+k \ne 15.$

So we count:

$\text{Required number} = \binom{15}{3} - (\text{number of triples with } i+j+k=15).$

3. Count triples of distinct positive integers

We must count distinct indices

$1 \le i<j<k\le 15$

such that

$i+j+k=15.$

Since the sum is only 15, automatically all values are within 1 to 15 if positive.

So we just find all positive integer triples with distinct entries and increasing order.

Let us list systematically:

• If $i=1$, then $j+k=14$ with $1<j<k$:

  $(1,2,12), (1,3,11), (1,4,10), (1,5,9), (1,6,8).$

• If $i=2$, then $j+k=13$ with $2<j<k$:

  $(2,3,10), (2,4,9), (2,5,8), (2,6,7).$

• If $i=3$, then $j+k=12$ with $3<j<k$:

  $(3,4,8), (3,5,7).$

• If $i=4$, then $j+k=11$ with $4<j<k$:

  $(4,5,6).$

Thus the total number of such triples is

$5+4+2+1=12.$

4. Subtract from total

Therefore,

$\text{Required number} = 455-12=443.$

5. Check options

• A: $12$  number excluded, not required answer
• B: $419$  incorrect
• C: $443$  correct
• D: $455$  ignores restriction

Hence the correct answer is

$\boxed{443}.$