permutation and combination

**panelopy123** · October 28th 2012, 02:22 AM

HOW many solutions are dere to the equation:
x1+x2+x3+x4+x5=21
where xi=1,2,3,4,5, is a non-negative integer such that

c)0<=x1<=10
d)0<=x1<=3,1<=x2<4,and x3>=15?

kindly give full soln,not simply ans

**Plato** · October 28th 2012, 05:48 AM

Originally Posted by panelopy123

HOW many solutions are dere to the equation:
x1+x2+x3+x4+x5=21
where xi=1,2,3,4,5, is a non-negative integer such that
c)0<=x1<=10
d)0<=x1<=3,1<=x2<4,and x3>=15?

These are done with generating functions.
d) Expand $\left( {\sum\limits_{k = 0}^{10} {x^k } } \right)\left( {\sum\limits_{k = 1}^4 {x^k } } \right)\left( {\sum\limits_{k = 15}^{20} {x^k } } \right)\left( {\sum\limits_{k = 0}^5 {x^k } } \right)^2$
The coefficient of $x^{21}$ answers your question.