Sure, we must first understand that the order in which the two buttons are pressed simultaneously does not matter, and they are considered as one event. Similarly, pressing the remaining button is another distinct event. Hence, we're dealing with a combinatorics problem.
Now, let's solve it step-by-step:
1. Firstly, we need to determine how many ways it is possible to press two buttons simultaneously out of a total of 5 buttons. This can be figured out by calculating combinations of 5 items taken 2 at a time. The formula for Combinations is:
C(n, r) = n!(r!(n-r)!)
Here, n is the total number of items, and r is the number of items to choose. The symbol "!" denotes factorial, which means the product of all positive integers less than or equal to the number itself.
So, number of ways to press 2 buttons simultaneously out of 5 = C(5, 2) = 5!/(2!(5-2)!) = 10.
Therefore, there are 10 different ways to simultaneously press two buttons out of five.
2. After this, out of the remaining 3 buttons, we need to choose one button to press individually. This can also be computed using the Combination formula:
Number of ways to choose 1 button out of 3 = C(3, 1) = 3!/(1!(3-1)!) = 3.
Hence, there are three different ways to choose one button from the remaining three.
3. Finally, the total number of combinations would be the product of these two individual combination quantities. This is because each way of pressing two buttons can be followed by any way of pressing one remaining button. Therefore, the overall number of different ways the task can be completed is:
Total combinations = 10 * 3 = 30.
To conclude, the total number of combinations to unlock the lock by pressing two buttons together and then one button individually is 30.