Suppose I had the set {1,2,3} and wanted to make a list of 5 of those elements that contained one element from the set and a pair of duplicates.  For example, (2, 1, 3, 3, 1).  The 2 is the single element and the pair of duplicates is 1,1 and 2,2. How many ways can this be done?

A monkey has filled in a 3 × 3 grid with the numbers 1, 2, . . . , 9. A cat writes down thethree numbers obtained by multiplying the numbers in each horizontal row. A dog writesdown the three numbers obtained by multiplying the numbers in each vertical column.Can the monkey fill in the grid in such a way that the cat and dog obtain the same listsof three numbers? What if the monkey writes the numbers 1, 2, . . . , 25 in a 5×5 grid? Or1, 2, . . . , 121 in a 11 × 11 grid? Can you find any conditions on n that guarantee that itis possible or any conditions that guarantee that it is impossible for the monkey to writethe numbers 1, 2, . . . , n^2 in an n × n grid so that the cat and the dog obtain the same lists of numbers? Please provide a mathematically rigorous solution with proofs that could be understood by a person at a high school level.

Suppose we want to choose   3 colors, without replacement, from the   4 colors red, blue, green, and purple.       How many ways can this be done, if the order of the choices does not matter?   How many ways can this be done, if the order of the choices does matter?

for every 2 minivans there are 5 other types of vehicles if there are 161 vehicles how many of them are not minivans?