Combination of Poker Hands from five cards in the GoCee.com Poker Center

Different poker hands that can be made from five cards:

C52₅ = 2,598,960

Four of a kind:

{C13₁ * C4₄} * C48₁ = 624

Full House:

{C13₁ * C4₃} * {C12₁ * C4₂} = 3,744

Three of a kind: (Trips)

{C13₁ * C4₃} * C48₂ = 58,656 {This calculation includes full houses.}
Less Full Houses
58,656 - 3,744 = 54,912

Two Pair:

{C13₁ * C4₂} * {C12₁ * C4₂} * C11₁ * C4₁ = 247,104
This calculation considers a hand such as 2-K's+2-Q's+1-A
to be a different hand than 2-Q's+2-K's+1-A.
The result must be divided by C2₁=2 to account for these duplicate hands.
247,104 / 2 = 123,552

One Pair:

* C4₂} * C48₃ = 1,349,088 {This calculation includes Full Houses and both ways of two pair}

Less Full House and both ways of two pair.

1,349,088- 3,744 - 247,104 = 1,098,240

Flush:

{C4₁ * C13₅} = 5,148

{This calculation includes straight flushes.}

Less straight flushes
5,148 - 40 = 5,108

Straight:

There are only 10 denominations that can be chosen
to be the highest card of a straight, these cards are the 5, 6, etc up to K, A.

* C4₁} * C4₁ * C4₁ * C4₁ * C4₁ = 10,240 {This calculation includes straight flushes}

Less straight flushes
10,240 - 40 = 10,200

Straight Flush:

{C10₁ * C4₁} = 40

{This calculation includes Royal Flushes}