Please show work where it can be shown. Thank you, much appreciated.

31. How many ways can a president and a vice president be determined in a club with twelve members?

36. How many counting numbers have four distinct nonzero digits such that the sum of the four digits is

(a) 10?

(b) 11?

42. In a 7/39 lottery, you select seven distinct numbers from the set 1-39, where order makes no difference. How many different ways can you make your selection?

51. At a wedding reception, the bride, the groom, and four attendants will form a reception line. How many ways can they be arranged in each of the following cases?

(a) any order will do

(b) The bridge and groom must be the last two in line

(c) The groom must be last in line with the bride next to him

57. How many of the possible 5-card hands from a standard 52-card deck would consist of the following cards?

(a) four clubs and one non-club

(b) two face cards and three non-face cards

(c) two red cards, two clubs, and a spade