Royal flush: A K Q J T in suit.

Chance of getting one, assuming you go for nothing else:

Getting one cold.
20/52 * 4/51 * 3/50 * 2/49 * 1/48
= 480 / 311875200
= 1 / 649740

Getting four cards and drawing the fifth.
20/52 * 4/51 * 3/50 * 2/49 * 47/48 * 5 * 1/47 [5C1]
= 22560 / 311875200 * 5 * 1 / 47
= 112800 / 14658134400
= 1 / 129948

Getting three cards and drawing the fourth and fifth.
20/52 * 4/51 * 3/50 * 47/49 * 46/48 * 10 * 2/47 * 1/46 [5C2]
= 518880 / 311875200 * 10 * 2 / 2162
= 10377600 / 674274182400
= 1 / 64974

Getting two cards and drawing the other three.  This is more
complicated, because a given hand can contain more than one two-card
royal flush subsets.
Two high cards, three low cards:
20/52 * 4/51 * 32/50 * 31/49 * 30/48 * 10 [5C2]
Two high cards in suit, a high card out of suit, two low cards:
20/52 * 4/51 * 15/50 * 32/49 * 31/48 * 10 * 3 [5C2] [3C1]
Two high cards in suit, one high in each of two other suits, one low:
20/52 * 4/51 * 15/50 * 10/49 * 32/48 * 10 * 3 [5C2] [3C1]
Two high cards in suit, one high in each other suit:
20/52 * 4/51 * 15/50 * 10/49 * 5/48 * 10 [5C2]
Two high cards in each of two suits, one low:
20/52 * 4/51 * 15/50 * 4/49 * 32/48 * 15 [5C2,2]
Two high cards in each of two suits, one high in another suit:
20/52 * 4/51 * 15/50 * 4/49 * 10/48 * 15 [5C2,2]
All of these need to be multiplied by 3/47 * 2/46 * 1/45.  This gives:
( 23808000 + 35712000 + 11520000 + 600000 + 2304000 + 720000 ) / 311875200
* 6 / 97290
= 447984000 / 30342338208000
= 61 / 4131582
= about 1 / 67730.85

Getting only one card and drawing the rest.  Again, we have to be
careful to count cases, and exclude those where the other four cards
include a two-card royal flush subset.
One high card, four low cards:
20/52 * 32/51 * 31/50 * 30/49 * 29/48 * 5 [5C1]
Two high cards in different suits, three low cards:
20/52 * 15/51 * 32/50 * 31/49 * 30/48 * 10 [5C2]
Three high cards in different suits, two low cards:
20/52 * 15/51 * 10/50 * 32/49 * 31/48 * 10 [5C2]
Four high cards (in different suits), one low card:
20/52 * 15/51 * 10/50 * 5/49 * 32/48 * 5 [5C1]
All of these need to be multiplied by 4/47 * 3/46 * 2/45 * 1/44.  This
gives us:
( 86304000 + 89280000 + 29760000 + 2400000 ) / 311875200 * 24 / 4280760
= 4985856000 / 1335062881152000
= 4328 / 1158908751
= about 1 / 267770

Getting no high cards and then drawing a royal flush.
32/52 * 31/51 * 30/50 * 29/49 * 28/48 * 20/47 * 4/46 * 3/45 * 2/44 * 1/43
= 11599257600 / 57407703889536000
= 7192 / 35595054495
= about 1 / 4949256.74

The total chance is thus
1 / 649740 + 1 / 129948 + 1 / 64974 + 61 / 4131582 + 4328 / 1158908751 +
  7192 / 35595054495
= ( 1 * 129948 * 64974 * 4131582 * 1158908751 * 35595054495 +
    649740 * 1 * 64974 * 4131582 * 1158908751 * 35595054495 +
    649740 * 129948 * 1 * 4131582 * 1158908751 * 35595054495 +
    649740 * 129948 * 64974 * 61 * 1158908751 * 35595054495 +
    649740 * 129948 * 64974 * 4131582 * 4328 * 35595054495 +
    649740 * 129948 * 64974 * 4131582 * 1158908751 * 7192 ) /
  ( 649740 * 129948 * 64974 * 4131582 * 1158908751 * 35595054495 )
= 40509251670641272105825315294688671200 /
    934983806298299536576727264429150499643200
= 4318151 / 99666152586
= about 1 / 23080.75



Flush: all five cards the same suit.

Chance of getting one, assuming you go for nothing else:

Getting one cold.
52/52 * 12/51 * 11/50 * 10/49 * 9/48
= 617760 / 311875200
= 33 / 16660
= about 1 / 504.85

Getting four cards and drawing the fifth.
52/52 * 12/51 * 11/50 * 10/49 * 39/48 * 5 * 9/47
= 13384800 / 311875200 * 9 / 47
= 120463200 / 14658134400
= 1287 / 156604
= about 1 / 121.68

Getting three cards and drawing the other two.
52/52 * 12/51 * 11/50 * 39/49 * 38/48 * 10 * 10/47 * 9/46
= 101724480 / 311875200 * 90 / 2162
= 9155203200 / 674274182400
= 24453 / 1800946
= about 1 / 73.65

Getting two cards and drawing the other three.
Getting A A B C D:
52/52 * 12/51 * 39/50 * 26/49 * 13/48 * 10 [5C2]
Getting A A B B C:
52/52 * 12/51 * 39/50 * 12/49 * 26/48 * 15 [5C2,2]
Both multiplied by 11/47 * 10/46 * 9/45
( 82255680 / 311875200 + 113892480 / 311875200 ) * 990 / 97290
= 194186678400 / 30342338208000
= 57629 / 9004730
= about 1 / 156.25101724480

There is no way to get only one card in a suit without getting at least
two in some other suit.

Total chance
33 / 16660 + 1287 / 156604 + 24453 / 1800946 + 1859 / 128639
= ( 33 * 156604 * 1800946 * 128639 +
    16660 * 1287 * 1800946 * 128639 +
    16660 * 156604 * 24453 * 128639 +
    16660 * 156604 * 1800946 * 1859 ) /
  ( 16660 * 156604 * 1800946 * 128639 )
= 23106505073589357728 / 604437212568492064160
= 172117 / 4502365
= about 1 in 26.16