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Niv-Mizzet Reborn is a card from a recent Magic: the Gathering set.

The card says:

When Niv-Mizzet Reborn enters the battlefield, reveal the top ten cards of your library. For each color pair, choose a card that’s exactly those colors from among them. Put the chosen cards into your hand and the rest on the bottom of your library in a random order.

The colour pairs mentioned are part of the cost of each card. Each card costs are made up of up to 5 different symbols (W, U, B, R, G). E.g. a card that costs WUB has 3 different colour pairs.

What I want to know is, from a deck of 60 cards, what are the probabilities of the top ten being up to 10 different pairs (e.g. the probabilities of 1 unique pair, 2 unique pairs etc) assuming you have 5 lots of each pair in the deck (e.g. 5×WU, 5×WB ...etc.) and 4 other cards (you need to have used 6 cards to play Niv-Mizzet reborn).

I've already tried to use www.deckulator.appspot.com to work or the probabilities of the perfect 10-pair draw.

But it looks like it's not as simple as just multiplying by the number of ways of drawing a combination of pairs (e.g. 10 choose 5 + 10 choose 6 etc). As those seems like too good a set of odds.