# Problem #12

There are many ways to indicate the order of operations in the expression a + b + c + d using parentheses, for example (a + b) + (c + d) or a + ((b + c) + d). In fact, it is well-known that if there are n terms, there are Binomial[2n-2,n-1]/n (these are known as Catalan numbers) ways of parenthesizing the expression. For four terms this yields 5 ways of parenthesizing. Of course, the associative law tells us that these all give the same result algebraically.

Consider the expression

a + b × c + d.
In this case, inserting parentheses to indicate the order of operations can result in algebraically distinct expressions. For example, (a + b) × (c + d) is not equal to a + ((b × c) + d). [Note that we are not using the traditional order of operations where multiplication takes priority over addition; the order must be explicitly indicated by parentheses.]

This month's problems:

• How many algebraically distinct expressions can be obtained from the expression above?
• What if we place four pairs of parentheses in
a + b × c + d × e + f?
• Six pairs of parentheses in
a + b × c + d × e + f × g + h?

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