Problem 8.8. A computer can apply three operations to a number: "increase by 2", "increase by 3", "multiply by 2". The computer was given the number 1 and was made to try all possible combinations of 6 operations (each of these combinations is applied to the initial number 1). After how many of these combinations will the computer end up with an even number?
Answer: 486.

Solution. If the number was odd before the last operation, then one of the two operations "increase by 3" or "multiply by 2" will make it even, but the third will not. Similarly, if the number was even before the last operation, then one of the two operations "increase by 2" or "multiply by 2" will make it even, but the third will not.

Thus, the first 5 operations can be performed in any order - this gives $3^{5}$ options. To get an even result on the last operation, there are exactly 2 options, so the total number of combinations is $3^{2} \cdot 2=486$.

## 9th grade