no of solns of x+y+z=15.My main is concern is how to solve it through multinomial theorem(Why and how it is used like that)please explain in detail

Here if 0 is allowed then x,y,z can take values from 0 to 15

so now we have to find coefficient of x¹⁵ in (1+x+x²......x¹⁵)(1+x+x²......x¹⁵)(1+x+x²......x¹⁵)

by solving with g.p. formula we get (1-x¹⁶)³(1-x)^-3

Here (1-x¹⁶)³ will have no term with x¹⁵ ,so we need to find the coefficient of x^15 in (1-x)^-3

We have coefficient of x^r in (1-x)^-n is given by ^n+r-1C_r-1 so we get coefficient of x^15 in (1-x)^-3 is ^3+15-1C_15-1 which is ¹⁷C₃.

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