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September 27th, 2018, 09:03 AM  #1 
Newbie Joined: Sep 2018 From: India Posts: 2 Thanks: 0  Permutations and Combinations Problem
There are n number of chocolates. We can pick either 1 or 3 chocolates at once. In how many ways can we pick all the chocolates? The n variable can have any value as 30,100,50 etc. 
September 27th, 2018, 09:47 AM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,986 Thanks: 995  
September 27th, 2018, 10:37 AM  #3 
Newbie Joined: Sep 2018 From: India Posts: 2 Thanks: 0 
If n=1: There can be one possible way. If n=2: There are two possible ways. One on the first pick, and another one on second. if n=3: There can be 4 combinations. Either all chocolates at once or one by one. I hope this gives an idea about the problem. 
September 27th, 2018, 11:33 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,324 Thanks: 1233 
The problem is given $n$ find $j, ~k \ni 3j + k = n$ fixing $j$ we see that $k = n  3j$ i.e. it is fixed for a given $j$ We also note that $0 \leq j \leq \left \lfloor \dfrac{n}{3} \right \rfloor$ So a given $n$ has $ \left \lfloor \dfrac{n}{3} \right \rfloor+1$ different combinations of 1's and 3's that sum to it. 

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