Simple Maths
Prompt
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).
Answer guidance
Correct answer: 4 Solution: By the associative property of multiplication, it doesn't help to insert parentheses that specify the order of multiplication. For example, the associative property tells us that $(2\cdot(3\cdot 4))\cdot (5+1)$ is the same as $2\cdot3\cdot4\cdot (5+1)$. So the only way to get different values is to group +1 with a different number of factors. We get \begin{align*} 2\cdot 3 \cdot 4 \cdot (5 + 1) &= 144, \\ 2\cdot 3 \cdot (4 \cdot 5 + 1) &= 126,\\ 2\cdot (3 \cdot 4 \cdot 5 + 1) &= 122, \\ (2\cdot 3 \cdot 4 \cdot 5) + 1 \hphantom{)} &= 121. \end{align*}In total there are $\boxed{4}$ possible values for the expression.
