Find the number of non negative integer solutions for the equation
Consider the situation like there are 10 similar looking (indistinguishable) marbles and 5 (indistinguishable) sticks which work as separators
A typical situation like
Represents the case where x1=2, x2=3, x3=1, x4=0, x5=2, x6=2
And
Represents the case when x1=0, x2=2, x3=0, x4=0, x5=3, x6=5
Represents the case when x1=2, x2=0, x3=2, x4=5, x5=1, x6=0
So, all such above arrangements represent a solution to the equation in whole numbers.
Hence the number of such solutions is equal to the number of such arrangements of 10 indistinguishable marbles, and 5 indistinguishable sticks.
The number of ways in which this can be done in is
.
Now, if we want the number of solutions for the equation
in natural numbers, we rewrite the equation as
Now each of
, with i=1, 2, 3, 4, 5, 6 is a whole number
Hence it is equivalent to the number of solutions to the equation
where 
ie.,
Hence the number of solutions for the equation
in natural numbers is
Consider the situation like there are 10 similar looking (indistinguishable) marbles and 5 (indistinguishable) sticks which work as separators
A typical situation like
Represents the case where x1=2, x2=3, x3=1, x4=0, x5=2, x6=2
And
Represents the case when x1=0, x2=2, x3=0, x4=0, x5=3, x6=5
Represents the case when x1=2, x2=0, x3=2, x4=5, x5=1, x6=0
So, all such above arrangements represent a solution to the equation
in whole numbers.Hence the number of such solutions is equal to the number of such arrangements of 10 indistinguishable marbles, and 5 indistinguishable sticks.
The number of ways in which this can be done in is
. Now, if we want the number of solutions for the equation
in natural numbers, we rewrite the equation as
Now each of
, with i=1, 2, 3, 4, 5, 6 is a whole numberHence it is equivalent to the number of solutions to the equation
where ie., Hence the number of solutions for the equation
in natural numbers is
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