In multinomial expansion of $(1+x^5+x^9)^{100}$, every term will be of form: $C(100 ; m, n, p) \times 1^m \times (x^5)^n \times (x^9)^p$, where m + n + p =100.
So, for the coefficient of $x^{23}$, (5*n + 9*p) should be equal to 23. There is only 1 pair exist for this condition to hold i.e (1,2).
So, (m, n, p) will be (97, 1, 2). Now, put these values in the term for $x^{23}$.
C(100; 97, 1, 2) =$\frac{100!}{(97! * 1! * 2!)}$ = 485100. This will be the coefficient of $x^{23}$.