By inserting parentheses, it is possible to give the expression 2 × 3 + 4 × 5 several values. How many different values can be obtained?

2009 AMC 12B Problem 6

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By inserting parentheses, it is possible to give the expression

2 \times 3 + 4 \times 5

several values. How many different values can be obtained?

2

3

4

5

6

The result depends on when the addition is performed relative to the two multiplications

Enumerate the distinct groupings; several parenthesizations collapse to the same value

The genuinely different groupings give

(2 \times 3) + (4 \times 5) = 26,

\ ((2 \times 3) + 4) \times 5 = 50,

\ 2 \times (3 + (4 \times 5)) = 46,

and

\ 2 \times (3 + 4) \times 5 = 70.

These are all distinct, so

4

values can be obtained.

Thus, the correct answer is C.