steiner triple system in design theory

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We have 𝑟 red points, 𝑏 blue points and some quantity of triples, each of them containing
2𝑟 + 𝑏 or 2𝑏 + 𝑟. Also we know that each two distinct point are contained in exactly one
triple.
(a) how many triples of the 2𝑟 + 𝑏 type? For each pair of distinct 𝑟 points there is only one
𝑟
𝑟(𝑟−1)
triple containing them. The quantity of these pairs is ( ) = 2 .
2
𝑏(𝑏−1)
𝑏
(b) the same way the quantity of 2𝑏 + 𝑟 triples is ( ) = 2 , i.e. there are
2
𝑏(𝑏 − 1) + 𝑟(𝑟 − 1)
2
triples at all.

From the other hand, each two points among all 𝑟 + 𝑏 have exactly one triple. This way...

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