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# Step Problem

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How many ways are there up an 11 step staircase if you can go one or two steps at a time?

Jul 10, 2018

#1
+22343
+1

How many ways are there up an 11 step staircase if you can go one or two steps at a time?

Let $$F_n$$ be the number of ways to climb n stairs taking only 1 or 2 steps.

1. Solution:

$$\small{ \begin{array}{|rcll|} \hline F_n &=& ^{n}C_0 +\ ^{(n-1)}C_1+\ ^{(n-2)}C_2+\ ^{(n-3)}C_3+\ ^{(n-4)}C_4+\ ^{(n-5)}C_5 + \ldots +\ ^{(n-k)}C_k\,\quad (n-k) \ge k \\\\ && \text{case of n=11 steps}: \\\\ F_{11} &=& ^{11}C_0 +\ ^{10}C_1+\ ^{9}C_2+\ ^{8}C_3+\ ^{7}C_4+\ ^{6}C_5 \\ &=& 1 +10+36+56+35+6 \\ &=& 144 \\ \hline \end{array} }$$

2. Solution:

$$\begin{array}{|rcll|} \hline F_n &=& \mathcal{F}_{n+1} \qquad \mathcal{F} \text{ is the Fibonacci number }\\\\ && \text{case of n=11 steps}: \\\\ F_{11} &=& \mathcal{F}_{12} \qquad \\ && \mathcal{F}_0 = 0 \\ && \mathcal{F}_1 = 1\\ && \mathcal{F}_2 = 1\\ && \mathcal{F}_3 = 2\\ && \mathcal{F}_4 = 3\\ && \mathcal{F}_5 = 4\\ && \mathcal{F}_6 = 5\\ && \mathcal{F}_7 = 13\\ && \mathcal{F}_8 = 21\\ && \mathcal{F}_9 = 34\\ && \mathcal{F}_{10} = 55\\ && \mathcal{F}_{11} = 89\\ && \mathcal{F}_{12} = 144\\ && \ldots \\ F_{11} &=& 144 \\ \hline \end{array}$$

Jul 11, 2018

#1
+22343
+1

How many ways are there up an 11 step staircase if you can go one or two steps at a time?

Let $$F_n$$ be the number of ways to climb n stairs taking only 1 or 2 steps.

1. Solution:

$$\small{ \begin{array}{|rcll|} \hline F_n &=& ^{n}C_0 +\ ^{(n-1)}C_1+\ ^{(n-2)}C_2+\ ^{(n-3)}C_3+\ ^{(n-4)}C_4+\ ^{(n-5)}C_5 + \ldots +\ ^{(n-k)}C_k\,\quad (n-k) \ge k \\\\ && \text{case of n=11 steps}: \\\\ F_{11} &=& ^{11}C_0 +\ ^{10}C_1+\ ^{9}C_2+\ ^{8}C_3+\ ^{7}C_4+\ ^{6}C_5 \\ &=& 1 +10+36+56+35+6 \\ &=& 144 \\ \hline \end{array} }$$

2. Solution:

$$\begin{array}{|rcll|} \hline F_n &=& \mathcal{F}_{n+1} \qquad \mathcal{F} \text{ is the Fibonacci number }\\\\ && \text{case of n=11 steps}: \\\\ F_{11} &=& \mathcal{F}_{12} \qquad \\ && \mathcal{F}_0 = 0 \\ && \mathcal{F}_1 = 1\\ && \mathcal{F}_2 = 1\\ && \mathcal{F}_3 = 2\\ && \mathcal{F}_4 = 3\\ && \mathcal{F}_5 = 4\\ && \mathcal{F}_6 = 5\\ && \mathcal{F}_7 = 13\\ && \mathcal{F}_8 = 21\\ && \mathcal{F}_9 = 34\\ && \mathcal{F}_{10} = 55\\ && \mathcal{F}_{11} = 89\\ && \mathcal{F}_{12} = 144\\ && \ldots \\ F_{11} &=& 144 \\ \hline \end{array}$$

heureka Jul 11, 2018