I am asking this question for concept clarification :-

How many integral solutions are there to a+b+c=18 when a $\displaystyle \ge$ 1 , b $\displaystyle \ge$ 2 , c $\displaystyle \ge$ 3 ?

Solution: Let u $\displaystyle \ge$ 0, v $\displaystyle \ge$ 0, w $\displaystyle \ge$ 0, then

a $\displaystyle \ge$ u+1 , b $\displaystyle \ge$ v+2 , c$\displaystyle \ge$ w+3,

Therefore, a + b + c = 18

or u+1 + v+2 + w+3 = 18

or u + v + w =12. From there we solve as usual.

My question is why are we using: Let u $\displaystyle \ge$ 0, v $\displaystyle \ge$ 0, w $\displaystyle \ge$ 0, then

a $\displaystyle \ge$ u+1 , b $\displaystyle \ge$ v+2 , c$\displaystyle \ge$ w+3

Is it to convert each variable i.e a,b and c to one unit each ( since a,b ,c are unequal) or for any other reason ? What is the underlying logic ? Please advise on the above.

Thanks in advance !