Your sample space consists of 10x six-faced dices, so there is a total of ${6^{10}} = {\text{60,466,176}}$ outcomes. The events you're interested in are 15, 11 55, 111 555, 1111 5555 or 11111 55555. (one, two, three, four or five _pairs_ of numbers).
Let the probability of getting exactly _one of such pair_ be $P\left( {one} \right)$, it can be written as the probability to get exactly one, five then any other values except those in the other 8 dice (2, 3, 4 or 6 = 4 options): $$P\left( {one} \right) = {4^8}$$
It follows, then: $$\eqalign{ & P\left( {two} \right) = {4^6} \cr & P(three) = {4^4} \cr & P\left( {four} \right) = {4^2} \cr & P(five) = 1 \cr} $$ Which ammounts to $$P = \frac{{{4^8} + {4^6} + {4^4} + {4^2} + 1}}{{{\text{60,466,176}}}} = \frac{{65,536 + 4,096 + 256 + 16 + 1}}{{{\text{60,466,176}}}}$$ $$P = \frac{{69,905}}{{60,466,176}} \approx 0.0012$$