Can anyone plz...help me in understanding the concept of non negative integral solution of
X1+X2+X3+.....Xn=m ???

Question

Can anyone plz...help me in understanding the concept of non negative integral solution of 
X1+X2+X3+.....Xn=m ???

parthkohli · Answer

Oh, this is great. I know a pretty good way to understand this. Hold on.

anonymous · Answer

pleaseb type full question as you see in your book

parthkohli · Answer

Imagine five stars like this:$$\large \star \star \star \star \star$$If we want to find the number of solutions to $x_1 + x_2 + x_3= 5$, we can first think about how to represent solutions graphically. The following is a graphical interpretation of a solution:$$\large \star \star \star |\star| \star $$This represents $3 + 1 + 1 = 5$. You can represent all solutions this way.

samigupta8 · Answer

This will turn out to be very lengthy method.....doing like dis vil help in getting the ans aftr a gr8 consumption of time...

parthkohli · Answer

So like$$\large | \star \star| \star \star \star$$The above represents $0 + 2 + 3 = 5$.

And no, we're not going to solve it like this. We're going to derive the expression$$\binom{m+n-1}{n-1}$$

samigupta8 · Answer

This is what i want to ask how do v get this formula ...??

parthkohli · Answer

That's what I'm doing. So do you get what I'm doing above? Like how I'm graphically representing the solutions

If you understand that we'll proceed with the derivation

samigupta8 · Answer

Yup....i got it...

parthkohli · Answer

OK, so finding the number of non-negative solutions to $x_1 +x_2+x_3 + \cdots + x_n = m$ is the same as the number of ways to separate $m$ stars by $n-1$ bars. 

Now the interesting part...

Think of $\large \star \star \star |\star| \star$ as a seven letter word. We have to choose two places for the bars and the rest of the places will be taken by the stars. So the number of solutions of $x_1 + x_2 + x_3 = 5$ is $\binom{7}2$.

parthkohli · Answer

Similarly, in general, we have $m$ stars and $n-1$ bars. Now we have an $m+n-1$ letter word out of which we will choose $n-1$ places for the bars. Therefore the number of solutions is$$\binom{m+n-1}{n-1}$$

parthkohli · Answer

Do you understand this approach?

samigupta8 · Answer

I m trying to understand it....:)

samigupta8 · Answer

U said that we need to place bars in between the letter words...ryt ...

parthkohli · Answer

Yeah, you can think of it that way.

samigupta8 · Answer

It is like dat way evn if we were to arrange the stars between the letter wrdd then also we wil get same result bcz then bars will automatically occupy the left over places...

parthkohli · Answer

$$x_1 + x_2 + x_3 +x_4= 7$$Seven stars, three bars. Total ten "letters". We have to choose three places for the bars. Total 10C3 solutions.