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mathmath333
 one year ago
Counting problem
mathmath333
 one year ago
Counting problem

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mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} & \normalsize \text{Find the number of natural number solutions of.}\hspace{.33em}\\~\\ & a+2b+c=100 \hspace{.33em}\\~\\ \end{align}}\)

dan815
 one year ago
Best ResponseYou've already chosen the best response.2b can be all even numbers

dan815
 one year ago
Best ResponseYou've already chosen the best response.2do you know how to solve the simpler version of this problem a+b+c=100

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1do i need to change 2b to d

dan815
 one year ago
Best ResponseYou've already chosen the best response.2sry i mean 2b is all even numbers

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1i meant to use substitution as 2b=d

dan815
 one year ago
Best ResponseYou've already chosen the best response.2here is another way u can think about it a+b+b+c = 100 how would you solve this a+b+c+d=100

dan815
 one year ago
Best ResponseYou've already chosen the best response.2i think you would solve a+b+c=100 and divide by 2

dan815
 one year ago
Best ResponseYou've already chosen the best response.2as all the odd possibilities would not possible

dan815
 one year ago
Best ResponseYou've already chosen the best response.2yeah but let me think that might not technically be true we dont know if odd and even solutions that add to 100 is exactly equal or not

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1How about calculatinb a+b1+b2+d=100 and substracting something

dan815
 one year ago
Best ResponseYou've already chosen the best response.2okay well maybe we can go back to the stars and bars method after this but i think we can do it with just sums

dan815
 one year ago
Best ResponseYou've already chosen the best response.2no wait there are 101 WAYs actually to add upto 100 when u pick 2b=0

dan815
 one year ago
Best ResponseYou've already chosen the best response.2does 0 count as natural number though?

Zarkon
 one year ago
Best ResponseYou've already chosen the best response.0sometimes...some define it to include 0 and others don't

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You could brute force it, write a python program to count it .

dan815
 one year ago
Best ResponseYou've already chosen the best response.2okay then lets subtract that out

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1No pen paper method i need

dan815
 one year ago
Best ResponseYou've already chosen the best response.2do you want to check with brute force

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1yea how do u brute force lol

dan815
 one year ago
Best ResponseYou've already chosen the best response.2tbh its just writing what i did up there in a code lol

dan815
 one year ago
Best ResponseYou've already chosen the best response.2i didnt really do it a very smart way summation is pretty bruteforcy

dan815
 one year ago
Best ResponseYou've already chosen the best response.2the answer is definately close to (99 choose 2)/2

dan815
 one year ago
Best ResponseYou've already chosen the best response.2because the odd + even combinations are not equal in this case there is going to be some fluctuation around that number

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1http://www.wolframalpha.com/input/?i=solve+a%2B2b%2Bc%3D100%2Ca%3E0%2Cb%3E0%2Cc%3E0+over+integers

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1a short research shows a+2b+c=10>4! ways a+2b+c=12>5! ways a+2b+c=14>6! ways Can i somehow get a pattern

dan815
 one year ago
Best ResponseYou've already chosen the best response.2@kainui can you see a way to do this problem combinatorically

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0this would be the 'brute force' method Python code: count=0 for a in range(1,100): for b in range(1,100): for c in range(1,100): if a + 2*b+c ==100: count = count + 1 print count >>> 2401

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1yea 2401 is correct

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0a+2b+c=10>4! ways a+2b+c=12>5! ways a+2b+c=14>6! ways you can derive a pattern from this

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} & \normalsize \text{When a and c are even then}\hspace{.33em}\\~\\ & 2a'+2b+2c'=100\hspace{.33em}\\~\\ & \implies a'+b+c'=50\hspace{.33em}\\~\\ & \implies \dbinom{49}{2}\hspace{.33em}\\~\\ & \normalsize \text{When a and c are odd then}\hspace{.33em}\\~\\ & 2a'1+2b+2c'1=100\hspace{.33em}\\~\\ & \implies a'+b+c'=51\hspace{.33em}\\~\\ & \implies \dbinom{50}{2}\hspace{.33em}\\~\\ & \normalsize \text{Total}=\dbinom{49}{2}+\dbinom{50}{2}=2401\hspace{.33em}\\~\\ \end{align}}\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1that pattern was incorrect

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0edit* actually the pattern should be a+2b+c=10>4^2 ways a+2b+c=12>5^2 ways a+2b+c=14>6^2 ways you can derive a pattern from this a+2b+c=n>(1/2*n 1)^2 ways

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0when n = 100, 1/2*100 1 = 49 and 49^2 = 2401

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thats a nice solution you have so how do you know that a' + b + c ' = 50 has (49 choose 2 ) ways to find positive solutions

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1u have to use stars and bars method https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0For any pair of positive integers n and k, the number of ktuples of positive integers whose sum is n is equal to the number of (k − 1)element subsets of a set with n − 1 elements. we had 50 is the sum , and 3 tuples so that means 49 choose 2

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0And then consider the case when a,b are both even or both odd. Note that a,c cannot be one even and one odd, because odd + even + even = odd

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1the language given is wikipdia is technical , if u have any doubt u can go through stars and bars videos on youtubes

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0oh, do you have a link? or should i google

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1here is one of the links https://brilliant.org/discussions/thread/starsandbars/ or simply try google or youtube
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