Straight in 52cards+2Joker

Question

I have 52card (ace to king) + 2Joker I'm supposed to compute how much straights of 5 cards I can make, excluding the straight flushes (straights with all cards being the same color)

My reasoning is :

• straight flush

  There is 9 * 4 = 36 straight flushes
  Cardinal( (1,2,3,4,5), (2,3,4,5,6), ..., (9, 10, J, Q, K) ) = 9 (or 10 if we add royal flushes)
  4 differents colors
  --> To exclude the straight flushes I only need to substract 36 to my result
  

• straight :

  5 cards in 4 colors generating 9 patterns :
  5^4 * 9  straights possible  
  

My intermediate result is :

(5^4 * 9) - 36

But it doesn't take in account the 2 jokers and I just can't find how to do so.

Answer 1

Adapt your method. First count all straight flushes that may use wild cards. Then subtract that from a count of straights.

For straight flushes:

1. the five cards are all genuine; choose the lowest rank from $\{A,2,\ldots,10\}$; choose the suit; $$\binom{10}{1}\binom{4}{1}=40$$

2. one wild is used; choose which wild; the wild could replace any of the upper four ranks, but not the lower one or why wouldn't you count it as being at the high end? this leaves out the straight flushes of the form $\{*,J,Q,K,A\}$; $$\binom{10}{1}\binom{4}{1}\binom{2}{1}\binom{4}{1}+8=328$$

3. two wilds are used; they could replace any two from the top four; there are similar upper-end considerations: $$\binom{10}{1}\binom{4}{1}\binom{4}{2}+16=256$$

So that makes $624$ straight flushes. Can you adapt this to count straights?