Given-3p-2-p-q-3-12-find-the-coefficient-of-p-10-q-6-

Question Number 204815 by cortano12 last updated on 28/Feb/24

Given {({3 p}^{2} - p + q^{3})}^{12}, find the

coefficient of p^{10} q^{6}

Answered by TonyCWX08 last updated on 28/Feb/24

\begin{array}{ccccc} 3 p^{2} & - p & q^{3} \\ 4 & 2 & 2 \\ 3 & 4 & 2 \\ 2 & 6 & 2 \\ 1 & 8 & 2 \end{array}

12 + 9 + 6 + 3

= 30

Answered by mr W last updated on 28/Feb/24

{(3 p^{2} - p + q^{3})}^{12}

= \sum_{a + b + c = 12} (\begin{matrix} 12 \\ a, b, c \end{matrix}) {(3 p^{2})}^{a} {(- p)}^{b} {(q^{3})}^{c}

3 c = 6 \Rightarrow c = 2

a + b = 12 - 2 = 10

2 a + b = 10

\Rightarrow a = 0, b = 10

c o e f . o f p^{10} q^{6} i s

(\begin{matrix} 12 \\ 0, 10, 2 \end{matrix}) 3^{0} {(- 1)}^{10} 1^{2} = \frac{12!}{10! 2!} = 66 ✓

Commented by mr W last updated on 28/Feb/24

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