It is given that $(1+x)(1+{x}^{2})(1+{x}^{3})(1+{x}^{4})......(1+{x}^{100})$ ,Find the coefficient of ${x}^{9}$ in the expansion. I saw a question in my textbook but i stuck in it. The question is:
It is given that $(1+x)(1+{x}^{2})(1+{x}^{3})(1+{x}^{4})......(1+{x}^{100})$
a-)Find the coefficient of ${x}^{9}$ in the expansion.
b-)Find the coefficient of ${x}^{28}$ in the expansion.
Apart from this question, i wonder that what would happen if the question were in the form of $(1+a_0x)(1+a_1{x}^{2})(1+a_2{x}^{3})(1+a_3{x}^{4})......(1+a_{99}{x}^{100})$ where $a_1....a_{99}$ are rational numbers.
a-)Find the coefficient of ${x}^{9}$ in the expansion.
b-)Find the coefficient of ${x}^{28}$ in the expansion.
I am open to hints ,shortcuts and full solution for both of these two questions.Thanks for your helps..

Hint : The coefficient of $x^9$ is the number of ways you can decompose $9$ as a sum of distincts positive integers.
