Q) Find the number of solutions in positive integers to:
FIRSTLY: Note that all would have odd solutions iff
This is actually trivial: as we have and we can add numbers an even number of times to get an number. and add them number of times to get odd numbers. in short: and in both cases
So, to the solution:
I don't know if this is possible to prove by multinomial theorem, but simple application of stars and bars(or in this case, the binary set of 's and 's) work:
I am going to do a brief explaining to those who aren't well aware of this method The idea is to create a set which has a correspondance with the solution set:
among the different arrangements,each type of arrangement of the numbers in the binary set corresponds to a solution.
Example: if we had to find the total number of positive integral solutions (indefinite as to odd or even) to
We would create a set with one's and zeroes as partitions. For a more visual explanation: one arrangement of our set may look like:
and more importantly: this set will correspond to a the solution triple
thus, different arrangements would correspond to different solution sets, and thus we are reduced to finding the number of different arrangements of the and in our binary element set.
Basically this is the stars and bars principle
In this method: we use as partitions between the numbers and each represents....well each . So, we have 's and .
Now, since the least odd positive integer is , we, at first separate because they would .
after addition of fixed , we have our set looking like:
By our previous arguements, is . SO:
We group the i.e :
Writing and noting that each corresponds to a numerical value of . and since we already have ones fixed for each variable, the addition of any number of or, in this case, would result in the formation of odd numbers ()
The required no. of solutions in odd positive integers for the given equation is
This is the permutation of a total objects, out of which: objects are of one kind(), and other objects of the same kind ().
This is equal to:
An example to remove all doubts:
Find the odd positive solutions in to the equation:
In This case: we have , . therefore, solutions are actually possible.
Next step, is to create the set with zeroes and ones:
next, write the ones as
We have to find the no. of arrangements of zeroes and 's
this is equal to:
We now see that these are the only solutions: