Question

What will be the coefficients of ${a^8}{b^4}{c^9}{d^9}$ in ${\left( {abc + abd + acd + bcd} \right)^{10}}$?

Answer 1

Hint: In this question use the direct formula for any general term for equation in form of ${\left( {abc + abd + acd + bcd} \right)^n}$which is $\dfrac{{n!}}{{x!.y!.z!.q!}}{\left( {abc} \right)^x}{\left( {abd} \right)^y}{\left( {acd} \right)^z}{\left( {bcd} \right)^q} = \dfrac{{n!}}{{x!.y!.z!.q!}}{a^{\left( {x + y + z} \right)}}{b^{\left( {x + y + q} \right)}}{c^{\left( {x + z + q} \right)}}{d^{\left( {y + z + q} \right)}}$.The direct power and coefficients comparison will get to the answer.

Complete Step-by-Step solution:
As we know the general term of ${\left( {abc + abd + acd + bcd} \right)^{10}}$ is
$\dfrac{{10!}}{{x!.y!.z!.q!}}{\left( {abc} \right)^x}{\left( {abd} \right)^y}{\left( {acd} \right)^z}{\left( {bcd} \right)^q} = \dfrac{{10!}}{{x!.y!.z!.q!}}{a^{\left( {x + y + z} \right)}}{b^{\left( {x + y + q} \right)}}{c^{\left( {x + z + q} \right)}}{d^{\left( {y + z + q} \right)}}$
Now we need the coefficient of ${a^8}{b^4}{c^9}{d^9}$
So on comparing this with above equation we have,
$ \Rightarrow x + y + z = 8$................... (1)
$ \Rightarrow x + y + q = 4$................... (2)
$ \Rightarrow x + z + q = 9$................... (3)
$ \Rightarrow y + z + q = 9$.................. (4)
Now add all the four equation we have,
$ \Rightarrow 3x + 3y + 3z + 3q = 8 + 4 + 9 + 9 = 30$
Now divide by 3 we have,
$ \Rightarrow x + y + z + q = 10$ ............................. (5)
Now subtract equation (1), (2), (3) and (4) from equation (5) respectively we have,
$ \Rightarrow x + y + z + q - x - y - z = 10 - 8$
$ \Rightarrow q = 2$
And
$ \Rightarrow x + y + z + q - x - y - q = 10 - 4$
$ \Rightarrow z = 6$
And
$ \Rightarrow x + y + z + q - x - z - q = 10 - 9$
$ \Rightarrow y = 1$
And
$ \Rightarrow x + y + z + q - y - z - q = 10 - 9$
$ \Rightarrow x = 1$
Therefore the coefficient of ${a^8}{b^4}{c^9}{d^9}$ is
$ \Rightarrow \dfrac{{10!}}{{1!.1!.6!.2!}}{a^8}{b^4}{c^9}{d^9}$
Now simplify the above equation we have,
$ \Rightarrow \dfrac{{10 \times 9 \times 8 \times 7 \times 6!}}{{1 \times 1 \times 6! \times 2 \times 1}}{a^8}{b^4}{c^9}{d^9}$
$ \Rightarrow \dfrac{{10 \times 9 \times 8 \times 7}}{2}{a^8}{b^4}{c^9}{d^9}$
$ \Rightarrow 2520{a^8}{b^4}{c^9}{d^9}$
So the required coefficient of ${a^8}{b^4}{c^9}{d^9}$ in ${\left( {abc + abd + acd + bcd} \right)^{10}}$ is 2520.
So this is the required answer.

Note: Such types of questions are direct formula based and it is always advised to remember these direct formulas. It’s not a binomial expansion so we need not to be confused between these two concepts. Both are different to each other, any general term in the binomial expansion of ${\left( {x + y} \right)^n}$ is its ${\left( {n - r + 2} \right)^{th}}$ term.