The number of ways in which 16 sovereigns can be distributed among three applicants such that each applicant does not receive less than 3 sovereigns, is

Let $x, y, z$ be the number of sovereigns received by the three applicants. Then 

$x\ge 3,y\ge 3,z\ge 3\text{ and }x+y+z=16$

Thus, the required number of ways is equal to the number of solutions of the equation $x+y+z=16,\text{ where }x\ge 3,y\ge 3$ $z\ge 3$

Clearly, when x and y take their minimum values, z will assume its maximum value. Therefore, the maximum value of z is given by 

3+ 3+z=16=>z=l0

Similarly, the maximum values of x and y are each equal to 10.

Thus, the required number of ways is same as the number of solutions of the equation $x + y + z = 16,$ where $3\le x,y,z\le 10$

Hence, required number of way 

= Coefficient of $x^{16}\mathrm{in}{\left(x^3+x^4+\dots +x^{10}\right)}^3$

 = Coefficient of $x^7\text{ in }{\left(1+x+\dots +x^7\right)}^3$

 = Coefficient of  $x^7\text{ in }{\left(\frac{1-x^8}{1-x}\right)}^3$

${=}^7+3-1C_{3-1}{=}^9{C}_2=36$