Question

From a standard 52 card deck, how many 6 card hands contain:

(a) three (different) pairs

(b) a five card straight and a pair (a straight can only begin with A, 2, 3, ..., 10)

(c) only a high card (ie. no pair, no five card straight, no five card flush)

Please answer all parts of the question with explanation

Answer 1

a)

There are ¹³C₃ ways to choose the denominations of the 3 pairs, and for each pair there are ⁴C₂ ways to choose the two suits,

Hence Possible Combinations = ¹³C₃ * ⁴C₂ = 61776

b)

If we order the 5-card hand from highest number to lowest, the first card may be one of the following: king, queen, jack, 10, 9, 8, 7, 6, or 5. (Note: the ace may be the card above a king or below a 2, but we would have a royal flush if it were the card above the king.) There are 9 possibilities. After the first card, whose suit we may choose in 4 ways, the remaining cards are completely determined.

Combinations = 9*⁴C₁ = 9*4

But, we want 5 straight cards, ie each card can be of any suit. So, we will have 9*4*4^4 = 9*4^5

Also, the 6^th card will form a fair with the remaining 5 cards.

Hence Overall Combinations = 9*4^5 * ⁵C₁ = 46080

c)

High card means we must avoid higher-ranking hands. All higher-ranked hands include a pair, a straight, or a flush.

Because the numbers showing on the cards must be six different numbers, we have ¹³C₆ choices for the six numbers showing on the cards. Each of the cards may have any of four suits

So, Possible Combinations = ¹³C₆ * (⁴C₁)⁶ = 7028736

We subtract the number of straights, flushes, and royal flushes. (Note that we avoided having any pairs or more of a kind.)

A straight (excluding straight-flush) is five cards showing consecutive numbers (but not all of the same suit)

Straights = 10*4⁶ = 40960

Flush {A flush (excluding straight-flush) is all cards the same suit (but not a straight)}:

flushes (not straight) = ⁴C₁* (¹³C₆ – 10) = 6824

Straight Flush = 9*4 = 36

A royal flush is ace, king, queen, jack, and ten of the same suit

Royal flushes = ⁴C₁ = 4

Hence,

Combinations = 7028736 – 40960 – 6824 – 36 – 4 = 6980912