Below is an equation that is not correct yet. By adding some plus signs and minus signs between the digits on the left side (without changes the order of the digits), the equation can be made correct. 123456789 = 100.

Question

Below is an equation that is not correct yet. By adding some plus signs and minus signs between the digits on the left side (without changes the order of the digits), the equation can be made correct. 123456789 = 100.
Question: How many different ways are there to make the equation correct? One way: 123 – 45 – 67 + 89 = 100. Can you give a systematic way to find answers ?123 + 45 –67 + 89 = 100, 1 + 2 – 3 + 4 + 5 + 6 + 78 + 9 = 100, 12 + 3 – 4 + 5 + 67 + 89 = 100, 1 + 2 + 34 – 5 + 67 – 8 + 9 = 100, 1 + 23 – 4 + 5 + 6 + 78 –9 = 100, 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100, 12 –3 – 4 + 5 – 6 + 7 + 89 = 100, 12 + 3 4 + 5 –6 – 7 + 89 = 100, 1 + 23 – 4 + 56 + 7 + 8 + 9 = 100

Philoid · Accepted Answer

There are 11 different way to solve this equation.

i. 123 + 45 –67 + 89 = 100,

ii. 1 + 2 – 3 + 4 + 5 + 6 + 78 + 9 = 100,

iii. 12 + 3 – 4 + 5 + 67 + 89 = 100,

iv. 1 + 2 + 34 – 5 + 67 – 8 + 9 = 100,

v. 1 + 23 – 4 + 5 + 6 + 78 –9 = 100,

vi. 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100,

vii. 12 –3 – 4 + 5 – 6 + 7 + 89 = 100,

viii. 12 + 3 4 + 5 –6 – 7 + 89 = 100,

ix. 1 + 23 – 4 + 56 + 7 + 8 + 9 = 100

x. 123 – 45 – 67 + 89 = 100

xi. 123 + 45 + 67–89 = 100