Combinations, permutations and poker hands.

1. Given a standard deck of 52 cars, how many ways are there to draw 5 cards
to obtain each hand.
(a) Royal Flush
[ace, king, queen, jack, ten in the same suit]
(b) Straight flush
[five cards in sequence and of the same suit, but not ace king queen jack ten]
(c) Four of a kind
[four cards of one face value and one other card]
(d) Full house
[3 cards of one face value and 2 cards of another face value]

3. (e) Flush
[5 cards of the same suit but not in sequence, not including the straight flush
and royal flush above]

4. (f) Straight
[5 cards in sequence, but not all of same suit. Ace high or low]

5. (g) Three of a Kind
[exactly 3 cards of one face value and 2 different cards]

6. (h) Two pairs
[one pair of each two different face values and a card of a third face value]

7. (i) One pair
[two of one face value, and 3 cards of different face values, no matching the
pair]

8. (j) No pairs
[5 different face values, not in sequence, not all cards in the same suit]

9. Colours
In how many ways can 4 balls be drawn together from a bag containg 5 red balls, 7 white balls, and 3 green balls? How many of these ways
include:
(a) 2 red and 2 white balls? (b) just 2 white balls?
(c) only white balls? (d) balls of only one colour?
(e) no red balls?
More Points
Ten points are marked in a plane, no three of them lying in a straight line. How many triangles do they determine?
Beads
From 10 beads of different colours, how many bracelets of 6 beads each can be formed?
Books
From 7 French and 9 English books, in how many