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math challenge II-A Combinatorics 7.8

 
 
Picture of Zeyin Wu
math challenge II-A Combinatorics 7.8
by Zeyin Wu - Tuesday, 28 September 2021, 3:41 PM
 

I don't quite understand why is that 1+ and 2+ needed, and where should they appear. The explanation in the book is also short.

Thanks.

 
Picture of John Lensmire
Re: math challenge II-A Combinatorics 7.8
by John Lensmire - Wednesday, 29 September 2021, 11:52 AM
 

I'll get you started with a few examples.

We have $F_3 = 2$. Sums to $3-1=2$ using $1$'s and $2$'s are: $$1+1, 2.$$

We have $F_4 = 3$. Sums to $4-1=3$ using $1$'s and $2$'s are: $$1+1+1, 1+2, 2+1.$$

We have $F_5 = 5$. Sums to $5-1=4$ using $1$'s and $2$'s are: $$1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2.$$

From here, you should then try writing out some additional examples to see if you can understand the pattern and how the next set of sums is related to the previous two.

Let us know if you have additional questions.