a rabbit climb up a flight of 10 stairs and can only hop 1 or 2 steps each time. he never hops down, only up. How many ways can the rabbit hop up the flight of 10 steps?
89
First lets look at the number of combinations (order won't matter) to go the 10 steps
Behind it I will then do the number of permutations
2 2 2 2 2 ---- 1
2 2 2 2 1 1 --- 6!/(4!2!) = 15
2 2 2 1 1 1 1 --- 7!/(3!4!) = 35
2 2 1 1 1 1 1 1 --- 8!/(2!6!) = 28
2 1 1 1 1 1 1 1 1 --- 9!/(1!8!) = 9
1 1 1 1 1 1 1 1 1 1 --- 1
total = 1+15+35+28+9+1 = 89
Combination
Number of ways
ten 1's
1
eight 1's, one 2
(
9
8
,
1
)
=
9
six 1's, two 2's
(
8
6
,
2
)
=
28
four 1's, three 2's
(
7
4
,
3
)
=
35
two 1's, four 2's
(
6
2
,
4
)
=
15
five 2's
1
Therefore, there are:
1
+
9
+
28
+
35
+
15
+
1
=
89
ways
use fibbonacchi sequence idiots
1,2,3,5,8,13,21,34,55,89
89 is the 10th number
smh
To find the number of ways the rabbit can hop up a flight of 10 stairs, you can use a concept from combinatorics called the Fibonacci sequence.
The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. It starts with 0 and 1, so the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
In this scenario, since the rabbit can only hop 1 or 2 steps at a time, it can be observed that the number of ways the rabbit can climb up a flight of 10 stairs is the 10th Fibonacci number.
Here is how you can calculate the 10th Fibonacci number:
1. Start with the first two numbers of the Fibonacci sequence, 0 and 1.
2. Add the two preceding numbers to get the next number in the sequence.
3. Repeat this process until you reach the 10th number.
Following this approach, you will find that the 10th Fibonacci number is 55. Hence, there are 55 different ways the rabbit can hop up the flight of 10 stairs.