Its right way?

[kangkan Bairagi]

The number of non-negative integer solutions of x1+x2+x3+x4=32

Ans:
32+4-1
C
4-1

 

[Deleted member 4993]

The number of non-negative integer solutions of x1+x2+x3+x4=32

Ans:
32+4-1
C
4-1

Are those answers that you have worked out? Please explain the answers (worked out by you).

Does x1, x2, x3 & x4 have to be different?

 

[Romsek]

Are those answers that you have worked out? Please explain the answers (worked out by you).

It's one answer. \(\displaystyle \dbinom{32+4-1}{4-1}\) i.e. they've used stars and bars to determine the number of 4 element non-negative integer partitions of 32.

I doubt OP came up with this on their own.

OP do you know what an integer partition is or why the answer given is correct?

 

[pka]

The number of non-negative integer solutions of x1+x2+x3+x4=32

Ans:
32+4-1
C
4-1

To kangkan Bairagi, can you explain why \(^{31}\mathcal{C}_3\) positive integer solutions to \(x_1+x_2+x_3+x_4=32~?\)

 

[kangkan Bairagi]

sir
i am not sure but explain that
we will never sum, element number with equal number for positive integer & can we 1 deduct for same rule this way.
if it is the wrong way, so plz describe me.
Thank you sir

 

[pka]

Suppose that we have seven variables: \(x_1+x_2+x_3+x_4+x_5+x_6+x_7=70\).
We can ask three questions:
a) how many non-negative integer solutions are there?
b) how many positive integer solutions are there?
c) a) integer solutions are there where \(x_1\ge 1,~x_2\ge 2~\&~x_3>3~?\)
All of these follow one counting idea: The number of ways to place \(N\) identical objects into \(K\) distinct cells is \(\dbinom{N+K-1}{N}\)
In part a) the 70 ones are identical & the seven x's are distinct. There \(\dbinom{70+7-1}{70}\) non-negative integer solutuons.
To solve part b), since a positive integer is at least one we take seven ones from the seventy and put one in each of the cells( the \(x's\)). Now we distribute the other \(63\) ones into the \(7~x's\) in \(\dbinom{63+7-1}{63}\) ways.
Now I ask you to use the idea in part b) to do part c).