The total number of 3- digit numbers, whose sum of digits is 10 is __________

Let abc be three digit number a+b+c=10 where a≥1and b,c≥0 Let a-1=x⇒a=x+1 and x≥0∴x+1+b+c=10⇒x+b+c=9 Total number of non‐negative solutions =C3-1   9+3-1=C2   11=55But in these solutions (10, 0, 0) is one of the solutions. Eliminate it, the anwer is 54.Alliter : The number of 3 digit numbers, with sum of digits of each number equals to 10 is given by the coefficient of x10  in (x+x2+x3+⋯⋯+x9) x (1+x+x2+⋯⋯+x9)2 = Coefficient of x9 in (1+x+x2+⋯+x8) x (1+x+x2+⋯+x9)= Coefficient of  x9 in 1-x91-x x 1-x101-x2 = Coefficient of x9 in 1-x9 x 1-x-2= Coefficient of x9 in 1-x9 x1 21.2 + 2.3.x + 3.4.x2 + ⋯ +10.11.x9 + ⋯⋯ ∞= 121.10.11 - 1.1.2  = 55 - 1 =54