The expression 2 x 3 x 4 x 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 x (3 x 4)) x (5+1) = 144.
In total, how many values can be obtained from the expression 2 x 3 x 4 x 5 + 1 by inserting parentheses? (Note that rearranging terms is not allowed, only inserting parentheses).
To determine the total number of values that can be obtained from the expression 2 x 3 x 4 x 5 + 1 by inserting parentheses, we need to consider the possible placements of parentheses.
We have four numbers to consider: 2, 3, 4, and 5. We can group them in different ways using parentheses.
Case 1: No numbers are grouped
In this case, the expression remains unchanged: 2 x 3 x 4 x 5 + 1.
Case 2: Two numbers are grouped
We can choose 2 numbers out of 4 to group together. The possible combinations are:
(2 3) x (4 5) + 1 = 24 x 20 + 1 = 481
(2 4) x (3 5) + 1 = 8 x 15 + 1 = 121
(2 5) x (3 4) + 1 = 10 x 12 + 1 = 121
(3 4) x (2 5) + 1 = 12 x 10 + 1 = 121
Case 3: Three numbers are grouped
We can choose 3 numbers out of 4 to group together. The possible combinations are:
((2 3) 4) x 5 + 1 = 24 x 5 + 1 = 121
((3 4) 2) x 5 + 1 = 14 x 5 + 1 = 71
((4 5) 2) x 3 + 1 = 18 x 3 + 1 = 55
((2 4) 3) x 5 + 1 = 12 x 5 + 1 = 61
Case 4: All numbers are grouped
We group all four numbers together: (((2 3) 4) 5) + 1 = 20 + 1 = 21
Therefore, by inserting parentheses, we can obtain the following values: 121, 144, 481, 71, 55, 61, and 21. In total, there are 7 values that can be obtained.
Please try again.
Apologies for the mistake in my previous response. Let's correct the calculation.
Using the rule of Order of Operations, the expression 2 x 3 x 4 x 5 + 1 should be evaluated as follows:
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 + 1 = 121
Now, let's consider the possible placements of parentheses and calculate the different values.
Case 1: No numbers are grouped
In this case, the expression remains unchanged: 2 x 3 x 4 x 5 + 1 = 121.
Case 2: Two numbers are grouped
We can choose 2 numbers out of 4 to group together. The possible combinations are:
(2 x 3) x (4 x 5) + 1 = 6 x 20 + 1 = 121
(2 x 4) x (3 x 5) + 1 = 8 x 15 + 1 = 121
(2 x 5) x (3 x 4) + 1 = 10 x 12 + 1 = 121
Case 3: Three numbers are grouped
We can choose 3 numbers out of 4 to group together. The possible combinations are:
((2 x 3) x 4) x 5 + 1 = 24 x 5 + 1 = 121
(2 x (3 x 4)) x 5 + 1 = 24 x 5 + 1 = 121
(2 x 3 x 4) x 5 + 1 = 24 x 5 + 1 = 121
(2 x 3) x (4 x 5) + 1 = 6 x 20 + 1 = 121
Case 4: All numbers are grouped
We group all four numbers together: ((2 x 3) x 4) x (5 + 1) = 24 x 6 = 144.
Therefore, by inserting parentheses, we can obtain the following values: 121 and 144. In total, there are 2 values that can be obtained.
Actually, it's WAY higher. Please try again.
Apologies for the confusion. Let's recalculate.
Using the rule of Order of Operations, the expression 2 x 3 x 4 x 5 + 1 should be evaluated as follows:
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 + 1 = 121
Now let's consider the possible placements of parentheses and calculate the different values.
Case 1: No numbers are grouped
In this case, the expression remains unchanged: 2 x 3 x 4 x 5 + 1 = 121.
Case 2: Two numbers are grouped
We can choose 2 numbers out of 4 to group together. The possible combinations are:
1. (2 x 3) x (4 x 5) + 1 = 6 x 20 + 1 = 121
2. (2 x (3 x 4)) x 5 + 1 = 2 x 12 x 5 + 1 = 121
3. (2 x 3 x 4) x 5 + 1 = 24 x 5 + 1 = 121
4. 2 x (3 x (4 x 5)) + 1 = 2 x 3 x 20 + 1 = 121
Case 3: Three numbers are grouped
We can choose 3 numbers out of 4 to group together. The possible combinations are:
1. (2 x 3) x (4 x 5) + 1 = 6 x 20 + 1 = 121
2. (2 x (3 x 4)) x 5 + 1 = 2 x 12 x 5 + 1 = 121
3. (2 x 3 x 4) x 5 + 1 = 24 x 5 + 1 = 121
4. 2 x ((3 x 4) x 5) + 1 = 2 x 60 + 1 = 121
5. 2 x (3 x (4 x 5)) + 1 = 2 x 3 x 20 + 1 = 121
6. (2 x 3 x 4 x 5) + 1 = 120 + 1 = 121
Case 4: Four numbers are grouped
We group all four numbers together:
1. ((2 x 3) x 4) x (5 + 1) = 24 x 6 = 144
Therefore, by inserting parentheses, we can obtain the following values: 121 and 144. In total, there are 2 values that can be obtained.