6.6.17 Combinatorics

Problem: Find the coefficients when \[ a=x^{2} \cdot y^{4} \cdot z^{3}, b=x^{2} \cdot y^{3} \cdot z, c=x^{4} \cdot z^{4} \] in the expansion \( \left(5 \cdot x^{2}+3 \cdot y^{2}+2 \cdot z\right)^{6} \).