# 1 Step, 2 Step Poster

1 Step, 2 Step Poster

## Problem

This poster is based on the problem 1 Step 2 Step.

The poster is available as a PDF, or the image below can be clicked on to enlarge it.

## Student Solutions

233 ways

**Why?**

__Method 1 - using shorter staircases__

Staircase with 1 step: 1 way

Staircase with 2 steps: 1, 1 or 2.

Staircase with 3 steps: 1+staircase with 2 steps $\rightarrow$ 2 ways, or 2+staircase with 1 step $\rightarrow$ 1 way, total 3 ways

Staircase with 4 steps: 1+staircase with 3 steps $\rightarrow$ 3 ways, or 2+staircase with 2 steps $\rightarrow$ 2 ways, total 5 ways

Staircase with 5 steps: 1+staircase with 4 steps $\rightarrow$ 5 ways, or 2+staircase with 3 steps $\rightarrow$ 3 ways, total 8 ways

Staircase with 6 steps: 8 + 5 = 13 ways

Staircase with 7 steps: 13 + 8 = 21 ways

Staircase with 6 steps: 21 + 13 = 34 ways

Staircase with 6 steps: 34 + 21 = 55 ways

Staircase with 6 steps: 55 + 34 = 89 ways

Staircase with 6 steps: 89 + 55 = 144 ways

Staircase with 6 steps: 144 + 89 = 233 ways

__Method 2 - using binomial coefficients__

Numbers of 1s and 2s that add up to 12

1 step |
2 steps |
Number of jumps |
Number of possible ways |

12 | 0 | 12 | 1 way - 1, 1, 1, ..., 1 |

10 | 1 | 11 | 11 ways - the 2 could be any one of the 11 jumps |

8 | 2 | 10 | 10 options for the 'first' 2, 9 options for the 'second' 2 - but we have to divide by 2 afterwards because the two 2s are interchangeable, there is no 'first' and 'second' 10$\times$9$\div$2 = 45 |

6 | 3 | 9 | 9 options for the 'first' 2, 8 for the 'second' and 7 for the 'third' - but now each option has been counted 3$\times$2$\times$1 times 9$\times$8$\times$7$\div$6 = 84 |

4 | 4 | 8 | 8$\times$7$\times$6$\times$5$\div$(4$\times$3$\times$2$\times$1 = 70 |

2 | 5 | 7 | Now use the options for the two 1s, to make it easier 7$\times$6$\div$2 = 21 |

0 | 6 | 6 | 1 way - 2, 2, 2, ... |

Total number of ways: 1 + 11 + 45 + 84 + 70 + 21 + 1 = 233