Online Course Discussion Forum

Math Challenge lll Winter 2023-2024 Combinatorics

 
 
Picture of Bella Xiao
Math Challenge lll Winter 2023-2024 Combinatorics
by Bella Xiao - Thursday, 14 December 2023, 7:09 AM
 

Hello, I'm having trouble on problems 2.15 and2.16 ,help would be appreciated.


 
Picture of John Lensmire
Re: Math Challenge lll Winter 2023-2024 Combinatorics
by John Lensmire - Thursday, 14 December 2023, 1:42 PM
 

Here are some hints:

For 2.15 (8 "+" signs and 6 "-" signs arranged with 5 sign changes):

Let's look at an example first: $\large{+--++++-+++---}$. Why does this have 5 sign changes? Well we have a plus, then some minuses (1st sign change), then some pluses (a 2nd sign change), etc. In other words, this example looks like $\large{(+)(-)(+)(-)(+)(-)}$, where, for example, (+) denotes some pluses. How many different ways could we arrange the +'s and -'s to get this? Are there other cases we need to consider?

For 2.16:

First off, try to answer this question first with some specific values of m, n, and k. For example, consider first something like m=6, n=5, and k=3. Then think about an idea similar to 2.15, where (G) denotes some green books and (B) some blue books. Once you do a few specific values hopefully the full pattern emerges!

Hope this helps! Let us know if you have any other questions.

Picture of Bella Xiao
Re: Math Challenge lll Winter 2023-2024 Combinatorics
by Bella Xiao - Tuesday, 19 December 2023, 4:14 AM
 
Thank you very much! I was able to solve these problems!!!