A) Find a recurrence relation for the number of ways to...

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one, two, or three stairs at a time. b) What are the initial conditions? c) In how many ways can this person climb a flight of eight stairs? A string that contains only $0 \mathrm{s}, 1 \mathrm{s},$ and 2 $\mathrm{s}$ is called a ternary string.

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one, two, or three stairs at a time.
b) What are the initial conditions?
c) In how many ways can this person climb a flight of eight stairs?
A string that contains only $0 \mathrm{s}, 1 \mathrm{s},$ and 2 $\mathrm{s}$ is called a ternary string.

Key Concepts

Recurrence Relations

Recurrence relations are equations that express each term of a sequence as a function of one or more of the preceding terms. They are particularly useful in solving combinatorial problems where a complex counting problem can be decomposed into simpler, similar subproblems. This concept provides a systematic way to build up the solution for larger cases from known solutions of smaller cases.

Initial Conditions

Initial conditions are the base values provided for the smallest instances of the problem in a recurrence relation. They are essential because they enable the computation of further terms of the sequence and guarantee a unique solution to the recurrence relation. In combinatorial problems, specifying the initial number of ways to achieve the simplest scenarios sets the stage for solving the general case.

Combinatorial Enumeration

Combinatorial enumeration involves counting the number of ways to construct an object or achieve a particular arrangement according to defined rules. This concept often uses recursive strategies or recurrence relations to break down a counting problem into manageable parts, allowing for the systematic exploration of all valid configurations.

Transcript

00:01 Okay, so for this problem, we'll let a .n.

00:04 Be the number of ways.

00:07 A person can climb the stairs, but the stairs can be taken one, two, or three steps at a time.

00:32 Okay, so we can break this down into three cases.

00:38 So case one is if the last stair, or the last step is taken by one step.

00:55 Okay, so last step was one stare in length.

01:03 Well, then we're going to have the number of ways you can climb this way, be the number of ways you can climb all the previous steps minus the one last step you took.

01:18 So that's going to give us a sub n minus 1.

01:25 The second case is if our last step is taken by two stairs at that time.

01:37 Okay, so the number of ways we can do this, right, if you imagine, we have a little staircase.

01:43 And we took two steps the last time.

01:48 Well, the number of ways you can do this are all the number of different ways that you could climb all the preceding steps, right? and that value is going to be a n minus 2.

02:02 You might see where case 3 is going here.

02:07 If our last step is taken three stairs.

02:23 Well, in that case, we'll have a, but n minus the last three stairs.

02:29 So the total number of ways that we can climb the stairs is going to be all the ways we can have our last step be one stair.

02:36 All the way is that it can be two stairs, and all the ways it can be three stairs added together.

02:41 So that gives us that it is going to be equal to a to n minus 1 plus a sub n minus 2 plus a sub n minus 3...

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