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I gather from several sources that the number of starting combinations of dealer up card and the 2 player hole cards is 550.

I was thinking along the lines that there are 10 distinct values. Of which the dealer can have any one. Conditioned on this value, this leaves 2 cards to be chosen from the remaining 9 values so using the classic binomial theorem result (9,2)=36.

Thus multiple 36*10 and you have 360 combinations.

Intuitively I can see I have not accounted for the fact that some of the distinct values occur more often than others but I can't seem to fix it mathematically?

Baz

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    That depends on what they define as a "combination." When you go to your own calculations, you say "remaining 9 values"... why not ten values to choose from? What is stopping you from having a same card as the opponent? Why choose 2? What is stopping you from having two of the same card? This isn't a ten card deck, this is a 52 card deck... If you were to count all faces as "the same" and all cards of the same rank as "the same" this ups the count to $10\cdot (\binom{10}{2}+10)=10\cdot (45+10)=550$ – 2017-02-28
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    If you actually want to consider an equiprobable sample space, I would recommend looking at all $52\cdot \binom{51}{2}=66300$ outcomes individually, letting rank and suit matter instead of ignoring them. Matters get worse and more confusing if you are using multiple decks in a [shoe](https://en.wikipedia.org/wiki/Shoe_(cards)), eventually making it easier to approximate by dealing with an infinite shoe to let all draws be independent of one another. – 2017-02-28
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    I can see where the 45+10 comes from, its the choice of 2 from 10 for the player plus one from 10 for the dealer. But why is this multiplied by an additional 10? – 2017-02-28
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    $\color{red}{10}\cdot (\color{blue}{\binom{10}{2}}+\color{green}{10})$, the red $10$ comes from the number of choices for the dealer's face up card. The blue $\binom{10}{2}$ comes from the number of ways the player can have two different cards. The green $10$ comes from the number of ways the player can have two cards of the same value. Adding the green to the blue, you get the number of possible unique hands the player can have. You multiply the number of hands the player can have to the number of face up cards the dealer can have to get the total number of starting configurations. – 2017-02-28
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    ah finally got it thank you – 2017-02-28

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