In an agricultural experiment, we wish to compare the yields of three different varieties of wheat. Call these varieties A, B and C. We have a ?eld that has been marked into a 3 * 3grid so that there are 9 plots available to plant the varieties. We randomly assign the varieties to the plots so that each variety appears 3 times in the grid.

a) How many different ways can the varieties be assigned to the plots?

b) The random assignment of varieties to plots is called a completely randomized design. A better design might be to randomly assign the varieties so that each variety appears once in every row of the grid. This is called a randomized block design. How many possible randomized block designs are there involving 3 varieties and a 3 * 3grid?

c) A third design would assign the varieties to the plots so that every variety appeared once in every row and once in every column of the grid. This is called a Latin square design. How many possible Latin square designs are there involving 3 varieties and a 3 * 3grid?

d) For r varieties and an r * r grid, how many designs of the ?rst two types are there? (The number of Latin Squares of a given oder is an open problem - solve it to become famous!!).

e) For the ?rst two designs (and the general case in d), what is the probability that at least one variety will appear in the same position in each row?