2009 AMC 10B Problem 7

By inserting parentheses, it is possible to give the expression

2\times3+4\times5

several values. How many different values can be obtained?

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Solution

There are five full parenthesizations of four numbers joined by three operations. Evaluating all five gives

((2\times3)+4)\times5=50

and

(2\times(3+4))\times5=70.

(2\times3)+(4\times5)=26.

2\times((3+4)\times5)=70

and

2\times(3+(4\times5))=46.

Thus the distinct values are

26, 46, 50,

and

70,

for a total of

4.

Thus, the correct answer is C.