x1+x2+x3+x4+x5=32
3 of the variable are naturals and even
2 of the variable are naturals and odd
nobody of the variables equals to 1 or 0
get the final solution
i think i need to use the inclusion exclusion principle
someone can help me with this please
Let be natural numbers.
Let
Then your equation can be expressed:
There are many solutions. To get one, just try some numbers. As long as they're natural, it doesn't matter.
Since none are 0 or 1, I'd just put all 2s until the last one, and then see what you need to make up 15.
Hence
Is a solution.
i dont understand why u deside that
2a=x1
2b=x2
2c=x3
2d+1=x4
2e+1=x5
you dont know wich of the variable is odd and wich is even,
you just know that there are 3 even and 2 odd
so i think that you cant decide that x4 and x5 are odd by give them the value 2d+1 and 2e+1
am i right? of not? why?
please give me more ditails...
thanks
It doesn't matter what you decide.
The question doesn't specify WHICH variables are odd or even, it just says that 3 are even, and 2 are odd. You are at liberty to decide which ones you want to make.
The fact of the matter is that there are hundreds of different solutions to the equation under the given requirements. It doesn't really matter what one you choose.
Mush’s solution is interesting and solves the problem if we specify that are the even variables and the others are odd. However, there are more solutions that Mush seems to indicate.
For example: gives which also works.
So we must count the number of solutions to in the positive integers (each variable at least 1).
That number is .
As has been pointed out, we do not know which of the variables the odd ones are.
So we to account for that consider that the two odds may be in places.
The total then is
Do you have a textbook that contains this material?
Placing N identical objects into k different cells can be done in ways.
However, some of the cells may be empty.
Like is a solution to in the nonnegative integers.
Placing N identical objects into k different cells, with no empty cell, can be done in ways.
The solutions to in the positive integers number .