5-card poker probabilities

(I do this as a warm up exercise when I need to think about combinatorics, weird as that may sound)

The total number of choices for 5 cards is

$$ 52 * 51 * 50 * 49 * 48 = 311875200 $$

One pair: 

$$ {13 \choose 1} * { 4 \choose 2} * { 12 \choose 3 } * { 4 \choose 1}^3 = 1098240 $$

Two pair:

$$ {13 \choose 2 } * { 4 \choose 2}^2 * { 11 \choose 1} * {4 \choose 1} = 123552 $$

Three of a kind: 

$$ {13 \choose 1} * {4 \choose 3} * { 12 \choose 2 } * { 4 \choose 1}^2 = 54912 $$

Straight: 

$$  {10 \choose 1} * { 4 \choose 1} ^ 5 - (straight flush,royal flush) =   {10 \choose 1} * { 4 \choose 1} ^ 5 - {10 \choose 1}* {4 \choose 1} = 10200 $$

Flush:

$$ {13 \choose 5} * {12 \choose 1} - (royal flush,straight flush) = {13 \choose 5} * {12 \choose 1} - {10 \choose 1}* {4 \choose 1} = 5108 $$

Full House: 

$$ {13 \choose 1} {4 \choose 3} * { 12 \choose 1} * { 4 \choose 2} = 3744 $$

Four of a kind:

$$ {13 \choose 1} * { 12 \choose 1} * { 4 \choose 1} = 624$$

Straight flush (not including royal flush):

Per suit there are 10 choices of straight (A - 5, 2 - 6, 3 - 7, 4 - 8, 5 - 9, 6 - 10, 7 - J, 8 - Q, 9 - K, 10 - A)

$$ {10 \choose 1} * {4 \choose 1} - {4 \choose 1} = 36 $$

Royal flush: 

$$ {4 \choose 1} = 4 $$