Plot the values of C8 r for r between 0 and 8 Why do the poi

Plot the values of C(8, r) for r between 0 and 8. Why do the points form a symmetric shape? 19) How many different five-card hands are there that contain a straight?^2 (A straight occurs when the five cards have five consecutive fact values. One example of a straight is 8-9-10-jack-queen. Note that the ace can be used as either the lowest card or the highest card in a straight.) 20) How many different eight-card hands are there with no more than three red cards? 21) What is the difference between versions 1 and 2 of the multinomial counting theorem?

Solution

i) C(n,r) = C(n,n-r)

So it has a symmetric shape

C(8,0) = C(8,8) = 1

C(8,1) = C(8,7) = 8

C(8,2) = C(8,6) = 28

C(8,3) = C(8,5) = 56

C(8,4) = 70

ii) there are 10 different types of straights(1to5,2to6, ......10toace). for each type we have 4^5 possible hands. since any card can be of any suit. So total number of different 5 cards hands are 4^5*10

iii) hands with zero red card + 1 red card + 2 red cards + 3red cards

C(26,8) + C(26,7)*C(26,1) + C(26,6)*C(26,2) + C(26,5)*C(26,3)