Question

No of solution of equation by pemutationa and combination

Answer 1

Hey.
Here is your answer.

Counting non-negative integral solutionsNumber of non-negative integral solutions of equation x1+x2+⋯+xn=kx1+x2+⋯+xn=k

= Number of ways in which k identical balls can be distributed into n distinct boxes

=(k+n−1 n−1) = (k+n-1)C(n-1)

Counting positive integral solutionsNumber of positive integral solutions of equation x1+x2+⋯+xn=kx1+x2+⋯+xn=k

= Number of ways in which kk identical balls can be distributed into nn distinct boxes where each box must contain at least one ball

=(k−1 n−1)  = (k-1)C(n-1)

So, Let x1,x2,⋯,xmx1,x2,⋯,xm be integers.

Then the number of solutions to the equation
x1+x2+⋯+xm=nx1+x2+⋯+xm=n
subject to the conditions a1≤x1≤b1,a2≤x2≤b2,a1≤x1≤b1,a2≤x2≤b2, ⋯,am≤xm≤bm⋯,am≤xm≤bm

is equal to the coefficient of xn in 

(x^a1+x^a1+1+⋯+x^b1)×(x^a2+x^a2+1+⋯+x^b2)⋯×(x^am+x^am+1+⋯+x^bm)

Thanks.