how many number of solutions to the equations a+b+c+d = 12 where 1<= a,b,c,d <=6?

a+b+c+d = 12 a, b, c and d must be at least 1 => So give 1 to each of the variables => a + b + c + d = 8 Number of non negative solutions = (8+4-1)C(4-1) = 11c3 = 165 But none of the variables can be more than 6. So, we have to remove certain cases. Give 6 more to a => a + b + c + d = 2 =>a is 7 which is wrong according to the given information. So, all these cases are to be removed. Number of non negative solutions = (2+4-1)C(4-1) = 5c3 => But 6 can be given to a, b, c or d So, the total number of cases to be removed = 4 * 5c3 = 4*10 = 40 Answer = 165 - 40 = 125