Number of integer solution of a linear equation
by fungarwai, Sep 15, 2018, 11:14 AM
Stars and bars
The number of positive integer solutions of
is 
It is equivalent to put m-1 bars between n stars which have n-1 gaps.
Integer with infinite range
Use generating function
to find the number of non-negative integer solutions of 
Find the least common factor![$L=[a_1,a_2,\dots,a_m]$](//latex.artofproblemsolving.com/d/b/0/db0160ecbde4187b7616a24b8d061c8f77e80811.png)
, the generating function can be expressed as
Integer with finite range
Roll m normal dices with numbers 1 to 6, find the probability that the sum of these numbers equals to n
Permutation of consecutive items
permutation of binary number with length L including/without k consecutive "1"
where the number of "1" and "0" are n and L-n respectively
Without k consecutive "1",
Including k consecutive "1",
The number of positive integer solutions of
is 
It is equivalent to put m-1 bars between n stars which have n-1 gaps.
,
is 
,Integer with infinite range
Use generating function
to find the number of non-negative integer solutions of 
Find the least common factor
![$L=[a_1,a_2,\dots,a_m]$](http://latex.artofproblemsolving.com/d/b/0/db0160ecbde4187b7616a24b8d061c8f77e80811.png)
, the generating function can be expressed as
and take number of solutions of 
![$[x^9]\frac{1}{(1-x)(1-x^2)(1-x^6)}=2\sum_{k=0}^1 \binom{k+1}{1}+\sum_{k=0}^0 \binom{k+1}{1}
=2\binom{3}{2}+\binom{2}{2}=7$](http://latex.artofproblemsolving.com/8/1/8/818fc13a5065dfd092dfcb8d2160dede2e0986a0.png)
Integer with finite range
Roll m normal dices with numbers 1 to 6, find the probability that the sum of these numbers equals to n
Permutation of consecutive items
permutation of binary number with length L including/without k consecutive "1"
where the number of "1" and "0" are n and L-n respectively
Without k consecutive "1",
Including k consecutive "1",
,
,
,
is 
This post has been edited 9 times. Last edited by fungarwai, May 14, 2022, 10:44 AM
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