When no variable may exceed 10,
The number of solutions = coefficient of x20 in (x+x2+x3+⋯+x10)4
= coefficient of x20 in [x(1−x10)1−x]4
= coefficient of x20 in x4(1−x10)4(1−x)−4
= coefficient of x20 in x4(1−4x10+6x20−⋯)(1−x)−4
Now,
Coefficient of x16 in (1−x)−4=19C3
Coefficient of x6 in (1−x)−4=9C3
Number of required solutions
=19C3−4⋅9C3=633
Alternatively,
x+y+z+w=20
Total number of positive solutions =20−1C4−1=19C3
Let one variable be greater than 10, say x.
Put x=x′+10, where x∈N
Then, x′+y+z+w=10
Number of positive solutions =10−1C4−1=9C3
Similarly, we can solve if other variables are greater than 10
Hence, required number of solutions =19C3−4⋅9C3