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Count ways to reach the n’th stair
There are n stairs. One person will go to 1'st to the n'th stair. Maximum how many stairs he/she can cross in one step is also given. With this information, we have to find possible ways to go to the n'th stairs.
Let us consider one can cross a maximum two stairs in each step. So we can find recursive relations to solve this problem. One can move to n'th stair, either from (n-1)'th stair or from (n-2)'th stair. So ways(n) = ways(n-1) + ways(n-2).
Input and Output
Input: The number of stairs, say 10 the maximum number of stairs that can be jumped in one step, say 2 Output: Enter number of stairs: 10 Enter max stair a person can climb: 2 Number of ways to reach: 89
Algorithm
stairClimpWays(stair, max)
Input − number of stairs, maximum stair jump in a single step.
Output − Number of possible ways to reach.
Begin define array count of size same as stair number count[0] := 1 count[0] := 1 for i := 2 to stair -1, do count[i] := 0 for j = 1 to i and j <= max; do count[i] := count[i] + count[i - j] done done return count[stair - 1] End
Example
#include<iostream>
using namespace std;
int stairClimbWays(int stair, int max) {
int count[stair]; //fill the result stair using bottom up manner
count[0] = 1; //when there are 0 or 1 stair, 1 way to climb
count[1] = 1;
for (int i=2; i<stair; i++) { //for stair 2 to higher
count[i] = 0;
for(int j=1; j<=max && j<=i; j++)
count[i] += count[i-j];
}
return count[stair-1];
}
int countWays(int stair, int max) { //person can climb 1,2,...max stairs at a time
return stairClimbWays(stair+1, max);
}
int main () {
int stair, max;
cout << "Enter number of stairs: "; cin >> stair;
cout << "Enter max stair a person can climb: "; cin >> max;
cout << "Number of ways to reach: " << countWays(stair, max);
}Output
Enter number of stairs: 10 Enter max stair a person can climb: 2 Number of ways to reach: 89
Updated on: 16-Jun-2020
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