In multinomial expansion of (1+x^5+x^9)^100, every term will be of form: C(100 ; m, n, p) * (1^m) * ((x^5)^n) * ((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) = 100!/(97! * 1! * 2!) = 485100. This will be the coefficient of x^23.