Question

\The total no. of integral solutions of equation \(x+y+z+t=29\) where \[x \ge 1, y \ge 2, z \ge 3,t \ge 0\]

Answer 1

Let \(x = a+1\), \(y = b+2\), \(z = c+3\) where \(a \ge 0, b\ge 0, c\ge0\). Now we have\[a+1+b+2+c+3+t=29\]\[\Rightarrow a + b + c + t = 23\]where all variables are nonnegative

Answer 2

\[\binom{23+4-1}{4-1}\]Accordingly, this is the answer.

Answer 3

yes

Answer 4

wowza that is hard

Answer 5

That was a clever trick to make all the variables the same like that, cool.

Answer 6

I will name my first born Parth