Here's my approach:
Each of the triplets receives 0 stickers: Then there are 13 stars and 2 bars so there \({15 \choose 2} = 105\) ways to distribute the stickers.
Each of the triplets receives 1 sticker: Then there are 10 stars and 2 bars for \({12 \choose 2} = 66\) ways to distribute the stickers.
Each of the triplets receives 2 stickers: Then there are 7 stars and 2 bars for \({9 \choose 2} = 36\) ways to distribute the stickers.
Each of the triplets receives 3 stickers: Then there are 4 stars and 2 bars for \({6 \choose 2} = 15\) ways to distribute the stickers.
Each of the triplets receives 4 stickers: Then there are 1 star and 2 bars for \({3 \choose 2} = 3\) ways to distribute the stickers.
So, there are \(105 + 66 + 36 + 15 + 3 = \color{brown}\boxed{225}\) ways to distribute the stickers.