Determine the total number of nono-negative integral solutions of `x_(1)+x_(2)+x_(3)+x_(4)=100`,

Options

A. `""^(100)C_(3)`
B. `""^(99)C_(3)`
C. `""^(100)C_(4)`
D. `""^(99)C_(4)`

Explanation

Here, the upper limit for the variables `x_(1),x_(2),x_(3)` is more than 100 and the lower limit is non-negative. Therefore, required number of solutions
= Coefficient of `x^(100)"in"(1+x+x^(2)+x^(3)+....)^(4)`
= Coefficient of `x^(100)"in"(1-x)^(-4)`
`=""^(100+4-1)C_(4-1)=""^(100)C_(3)`
ALTER Total number of non-negative integral solutions of the given equations is same as the number of ways of distributing 100 items among 4 peresons such that each person can recevie any number of items.
Hence, total number of solutions `=""^(100+4-1)C_(4-1)=""^(103)C_(3)`