Setp 1

Binomial theorem

\(\displaystyle{\left({x}+{y}\right)}^{{{n}}}=\sum^{{{n}}}_{\left\lbrace{j}={0}\right\rbrace}\) nj \(\displaystyle{x}^{{{n}-{j}}}{y}^{{{j}}}\)

We are interested in the term \(\displaystyle{x}^{{{8}}}{y}^{{{9}}}\in{P}{S}{K}{\left({3}{x}+{2}{y}\right)}^{{{17}}}\)

n=17

j=9

The coresponding term is then:

\(\displaystyle{\left({n},{j}\right)}{\left({3}{x}\right)}^{{{n}-{j}}}{\left({2}{y}\right)}^{{{j}}}={\left({17},{9}\right)}{\left({3}{x}\right)}^{{{17}-{9}}}{\left({2}{y}\right)}^{{{9}}}\)

\(\displaystyle={\frac{{{17}!}}{{{9}!{\left({17}-{9}!\right)}}}}{\left({3}{x}\right)}^{{{8}}}{\left({2}{y}\right)}^{{{9}}}\)

\(\displaystyle={\frac{{{17}!}}{{{9}!{8}!}}}{3}^{{{8}}}{x}^{{{8}}}{2}^{{{9}}}{y}^{{{9}}}\)

\(\displaystyle={24310}\cdot{3}^{{{8}}}{2}^{{{9}}}{x}^{{{8}}}{y}^{{{9}}}\)

\(\displaystyle={81},{662},{929},{920}{x}^{{{8}}}{y}^{{{9}}}\)

Thus the coefficient of \(\displaystyle{x}^{{{8}}}{y}^{{{9}}}\)is then 81, 662, 929, 920

So,

81, 662, 929, 920

Binomial theorem

\(\displaystyle{\left({x}+{y}\right)}^{{{n}}}=\sum^{{{n}}}_{\left\lbrace{j}={0}\right\rbrace}\) nj \(\displaystyle{x}^{{{n}-{j}}}{y}^{{{j}}}\)

We are interested in the term \(\displaystyle{x}^{{{8}}}{y}^{{{9}}}\in{P}{S}{K}{\left({3}{x}+{2}{y}\right)}^{{{17}}}\)

n=17

j=9

The coresponding term is then:

\(\displaystyle{\left({n},{j}\right)}{\left({3}{x}\right)}^{{{n}-{j}}}{\left({2}{y}\right)}^{{{j}}}={\left({17},{9}\right)}{\left({3}{x}\right)}^{{{17}-{9}}}{\left({2}{y}\right)}^{{{9}}}\)

\(\displaystyle={\frac{{{17}!}}{{{9}!{\left({17}-{9}!\right)}}}}{\left({3}{x}\right)}^{{{8}}}{\left({2}{y}\right)}^{{{9}}}\)

\(\displaystyle={\frac{{{17}!}}{{{9}!{8}!}}}{3}^{{{8}}}{x}^{{{8}}}{2}^{{{9}}}{y}^{{{9}}}\)

\(\displaystyle={24310}\cdot{3}^{{{8}}}{2}^{{{9}}}{x}^{{{8}}}{y}^{{{9}}}\)

\(\displaystyle={81},{662},{929},{920}{x}^{{{8}}}{y}^{{{9}}}\)

Thus the coefficient of \(\displaystyle{x}^{{{8}}}{y}^{{{9}}}\)is then 81, 662, 929, 920

So,

81, 662, 929, 920