A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time or any combination of 1s and 2s. The total number of ways in which the person can go up the stairs is

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A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1s and 2s. Let $x + 2 y = 10$ , where $x$ is the number of times he takes single steps and $y$ is the number of times he takes two steps. Cases for the total number of ways:

$I : Wh e n x = 0 an d y = 5 \frac{5 !}{5 !} II : Wh e n x = 2 an d y = 4 \frac{6 !}{2 ! \cdot 4 !} III : Wh e n x = 4 an d y = 3 \frac{7 !}{4 ! \cdot 3 !} IV : Wh e n x = 6 an d y = 2 \frac{8 !}{6 ! \cdot 2 !} V : Wh e n x = 8 an d y = 1 \frac{9 !}{8 ! \cdot 1 !} VI : Wh e n x = 10 an d y = 0 \frac{10 !}{10 ! \cdot 0 !} = 1 (2, 2, 2, 2, 2) = 15 (1, 1, 2, 2, 2, 2) = 35 (1, 1, 1, 1, 2, 2, 2) = 28 (1, 1, 1, 1, 1, 1, 2, 2) = 9 (1, 1, 1, 1, 1, 1, 1, 1, 2) = 1 (1, 1, 1, 1, 1, 1, 1, 1, 1, 1)$

Hence, the total number of ways is $1 + 15 + 35 + 28 + 9 + 1 = 89$ .

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