How many integral solutions are there to x+y+z+t=29, when `xge1,y gt 1, z ge3 and t ge0`?

Options

A. `""^(26)C_(4)`
B. `""^(26)C_(3)`
C. `""^(22)C_(3)`
D. `""^(29)C_(3)`

Explanation

Here, upper limit of all the variables is more than 29 and the limit is non-negative. So, the upper limit of each varible can be taken as infinitely.
= Coefficient of `x^(29)" in "(x^(1)+x^(2)+x^(3)+....)(x^(2)+x^(3)+....)(x^(3)+x^(4)+....)(x^(0)+x+x^(4)+....)`
= Coefficient of `x^(29)" in "x^(6)(1+x+x^(2)+....)^(4)`
= Coefficient of `x^(23)"in"(1-x)^(-4)`
`""^(23+4-1)C_(4-1)=""^(26)C_(3)`
ALITER We have, `xge1,yge2,zge3,andtge0`
Let `u=x-1,v=y-2,w=z-3,"Then "xge1impliesuge0,yge2impliesvge-0,zge3implieswge0`
Thus, we have `u+1+v+2+w+3+t=29,u+v+w+t+23`,
where `uge0,vge0,wge0,tge0`
The total number of solutions of this equation is `""^(23+4-1)C_(4-1)=""^(26)C_(3)`