Whenever we have upper and lower bounds and the distribution of identical objects is done in distinct boxes/entities then the concept of generating function and multinomial theorem comes into picture..Hence we can write as :
No of solutions of the equation : x1 + x2 + x3 = 10 where 2 <= x1,x2,x3 <= 4
So it is nothing but the coefficient of x10 in (x2 + x3 + x4)3 = x6 (1 + x + x2)3
==> coefficient of x4 in (1 + x + x2)3 = (1 - x3)3 (1 - x)-3
Now we know :
Coefficient of xr in (1 - x)-n = n-1+rCr
So we will get relevant coefficients using "1" and "-3x3 " term of (1 - x3)3 only as other exceed x4 ..
==> coefficient of x4 in (1 + x + x2)3 = 1 * (3-1+4C4 ) - 3 * 3-1+1C1
= 15 - 9
= 6
Hence required number of ways of distribution = 6