Bi-nominal expansion of 3 terms

Question

Find the coefficient of $x^{17}$ in the expansion of $(3x^7 + 2x^5 -1)^{20}$

I'm stuck in handling this question as I do know how to solve it when it has 2 terms.
But now it has 3.

I have no idea where to begin...

Answer 1

=the coefficient of $x^{17}$ in $(2x^5-1)^{20}$

$+\binom{20}13^1\cdot$ the coefficient of $x^{17-7}$ in $(2x^5-1)^{20-1}$

$+\binom{20}23^2\cdot$ the coefficient of $x^{17-2\cdot7}$ in $(2x^5-1)^{20-2}$

Answer 2

We can apply the binomial theorem twice in order to find the coeffcient. It is convenient to use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ of an expression.

We obtain \begin{align*} \color{blue}{[x^{17}]}&\color{blue}{(3x^7+2x^5-1)^{20}}\\ &=[x^{17}]\sum_{k=0}^{20}\binom{20}{k}(3x^7)^k(2x^5-1)^{20-k}\tag{1}\\ &=\binom{20}{0}3^0[x^{17}](2x^5-1)^{20}+\binom{20}{1}3^1[x^{10}](2x^5-1)^{19}\\ &\qquad +\binom{20}{2}3^2[x^3](2x^5-1)^{18}\tag{2}\\ &=3\binom{20}{1}[x^{10}]\sum_{j=0}^{19}\binom{19}{j}(2x^5)^j(-1)^{19-j}\tag{3}\\ &=-3\binom{20}{1}\binom{19}{2}2^2\tag{4}\\ &\,\,\color{blue}{=-41\,040} \end{align*}

Comment:

• In (1) we apply the binomial theorem once to $(3x^7+(2x^5-1))^{20}$.

• In (2) we observe that the term $(3x^7)^k$ contributes to $[x^{17}]$ only when $k=0,1,2$. We also apply the rule $[x^p]x^qA(x)=[x^{p-q}]A(x)$.

• In (3) we note that expanding $(2x^5-1)^n$ gives only exponents of $x$ which are a multiple of $5$. We can therefore stick to the middle summand with $[x^{10}]$ and we apply the binomial theorem again.

• In (4) we select the coefficient of $x^j$ accordingly.

Answer 3

The trinomial expansion: $$(a+b+c)^n=\sum_{\ \ \ \ \ i,j,k\\ i+j+k=n}{n\choose {i,j,k}}a^ib^jc^k,$$ where ${n\choose {i,j,k}}=\frac{n!}{i!j!k!}$.

If applied to the given trinomial: $$(3x^7 + 2x^5 -1)^{20}=\sum_{\ \ \ \ \ i,j,k\\ i+j+k=20}{20\choose {i,j,k}}(3x^7)^i(2x^5)^j(-1)^k.$$ The exponent of $x$ will be $17$ when: $$7i+5j=17 \Rightarrow (i,j,k)=(1,2,17).$$ Hence: $${20\choose {1,2,17}}3^12^2(-1)^{17}=\frac{20!}{1!2!17!}(-12)=-41040.$$