10. Consider the polynomial $p(x)=\left(1+x^{3^{1}}\right)\left(1+x^{3^{2}}\right)\left(1+x^{3^{3}}\right)\left(1+x^{3^{4}}\right)\left(1+x^{3^{5}}\right)\left(1+x^{39}\right)$, and suppose we expand the product, obtaining an expression of the form $a_{0}+a_{1} x+a_{2} x^{2}+$ $\ldots+a_{402} x^{402}$, where for example $a_{0}=a_{402}=1$. How many of the coefficients $a_{0}, \ldots, a_{402}$ are non-zero?
(A) 52
(B) 56
(C) 60
(D) 64
(E) 376
10. The answer is $(\mathbf{C})$. First, observe that the non-zero coefficients are at most $2^{6}=64$, depending on whether we take the term 1 or the term in $x$ in each factor. However, we can observe that $x^{3} \cdot x^{9} \cdot x^{27}=x^{39}$, so there are terms in the expansion of the product that contribute to a single term in the resulting polynomial. How many of these repeated terms (which we will then subtract from the 64 indicated above) are there? There are 4, namely $x^{39}, x^{39} \cdot x^{3^{4}}, x^{39} \cdot x^{3^{5}}$ and $x^{39} \cdot x^{3^{4}} \cdot x^{3^{5}}$.