. Let A be an n-element set. Suppose we have three types of labels to assign to the

Chapter 17, Problem 17.29

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. Let A be an n-element set. Suppose we have three types of labels to assign to the elements of A. We can call these labels good, bad, and ugly or give them less interesting names such as Type 1, Type 2, and Type 3. Let a; b; c 2 N. Define the symbol n a b c to be the number of ways to label the elements of an n-element set with three types of labels in which we give exactly a of the elements labels of Type 1, b of the elements labels of Type 2, and c of the elements labels of Type 3. Evaluate the following from first principles: a. 3 1 1 1 . b. 10 1 2 5 . c. 5 0 5 0 . d. 10 7 3 0 . e. 10 5 2 3 10 2 3 5 .