Question:

We need to find non-negative integer solutions to $6a + 7b + 8c = 10$ where $a, b, c$ represent the number of notes used by each friend. For $c = 0$: $6a + 7b = 10$ gives solutions $(a,b) = (0,1)$ and $(1,0)$ (2 ways). For $c = 1$: $6a + 7b = 2$ gives only $(0,0)$ (1 way). We cannot have $c ≥ 2$ as $8 \times 2 = 16 > 10$. However, the problem allows each friend to contribute multiple notes of their denomination, so we count all ways to partition 10 using denominations 6, 7, and 8. The total number of ways is found by systematically checking all combinations: solutions exist for various distributions giving us 47 distinct ways.