Problem number 12 states, Ahmad walks up 8 steps, going up either 1 or 2 steps at a time. There is a hole on the 6th step, so he cannot use this step. And how many different ways can Ahmad reach the top step? So here we have Ahmad at the bottom of the 8-step staircase, with a broken 6th stair. Let's quickly analyze what happens around this 6th step. When Ahmad is on the 5th step, he can either step on the 6th step or the 7th step, but he cannot on the 6th step because the 6th step is broken, so he must go to the 7th step. At the bottom of the 8-step staircase, there is a broken 6th stair. At the 7th step, Ahmad can only jump to the 8th step because there is no 9th step, so again, his move is forced and he must step on the 8th step. And now he is at the top. This means that once he gets to the 5th step, no more decisions can be made and he only has one path to get to the 8th step. This means that the number of ways to get to the 5th step is the number of ways to get to the 9th step. The number of ways to get to the 5th step is the number of ways to get to the 8th step. Knowing this, let's only consider the first 5 steps. One way to get to step 5 is to simply take 5 single steps. So he would go up 1, up 1, up 1, up 1, up 1. Another way might be to take 3 single steps and 1 double step. There are 4 possible positions that this double step could take place. It could be the 1st, the 2nd, or the 3rd, or the 4th step for Ahmad. This means that we have 4 additional ways to get to the 5th step. Now let's consider taking 1 single step and 2 double steps. There are 3 combinations here depending on where the single step is. It could be the 1st, 2nd, or the 3rd step. Those are the only combinations possible because 3 double steps is impossible because that would get us to step 6. So here we have 8 possible ways to get to the 5th step. And because the 5th step through the 8th step, Ahmad's moves are forged, there are 8 ways to get to the 8th step. So the question asks us, in how many different ways can Ahmad reach the top step? The answer is 8. Letter C.

staircase

broken step

climbing paths

8 ways

Ahmad