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(Solved): Problem 3: Reach for the Stairs [10 points] The staircase depicted below has 10 steps. As we walk ...
![Problem 3: Reach for the Stairs [10 points]
The staircase depicted below has 10 steps. As we walk up the staircase we must al](https://media.cheggcdn.com/media/561/561cac0e-1332-4b8f-b1fc-76a79d1f6a66/php0MiWyI)
Problem 3: Reach for the Stairs [10 points] The staircase depicted below has 10 steps. As we walk up the staircase we must always end on step 10. 10 1 2 3 5 6 7 8 9 A staircase with 10 steps
1. How many paths can I take up the staircase? Each path is distinguished by which steps touch. For example, one path is Start ? S?? S5 ? S6 ? S9 S10. I'm allowed to step over as many steps as I wish with each step - as long as I end on step 10. Each step goes UP the staircase, never down. 2. How many ways can I take up the staircase if I step on exactly 4 steps (including step 10). With each step, I must advance at least one step forward - I can not take a step and remain on the same step. 3. OPTIONAL (5 points extra credit): How many paths up the stairs are there if with each step I'm allowed to skip at most one level. For example: Start ? S? ? S3 S4 S5 ? S7 ? S9 S10 is one such path.
Expert Answer
SOLUTION: 1. In all cases, we have to touch Step 10. Hence our choice is limited to selection of any number of steps out of balance 9 numbers of steps, e.g. Step 1 to Step 9. We can select 0 step out of 9 available steps in 9C0 ways. Similarly, we ca