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The expression $$2\cdot 3 \cdot 4\cdot 5+1$$ is equal to 121, since multiplication is carried out before addition. However, we can obtain values other than 121 for this expression if we are allowed to change it by inserting parentheses. For example, we can obtain 144 by writing $$(2\cdot (3\cdot 4)) \cdot (5+1) = 144.$$ In total, how many values can be obtained from the expression ​​​​$$2\cdot 3 \cdot 4\cdot 5+1$$ by inserting parentheses? (Note that rearranging terms is not allowed, only inserting parentheses).

Aug 14, 2019

$$\text{Parens won't affect any of the multiplications on the left so the only way you can change\\ things is by changing the grouping that includes the addition}\\ (2\cdot 3\cdot 4\cdot 5+1)=121\\ 2\cdot (3\cdot 4\cdot 5+1)=122\\ 2\cdot 3\cdot (4\cdot 5+1)=126\\ 2\cdot 3\cdot 4\cdot (5+1)=132\\ \text{4 different values}$$