Question:
The coefficient of \(x^{12}\) in the expansion of
\[
(x^2+x+2)^8
\]
is
The coefficient of \(x^{12}\) in the expansion of
\[
(x^2+x+2)^8
\]
is
Show Hint
For multinomial expansions, first write the general term and equate the required power of the variable. Then find all valid non-negative integer solutions and add their contributions.
Updated On: Jul 29, 2026
- \(518\)
- \(448\)
- \(406\)
- \(182\)
Show Solution
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The Correct Option is A
Solution and Explanation
Concept:
In the multinomial expansion of
\[
(x^2+x+2)^8,
\]
a general term is
\[
\frac{8!}{r!\,s!\,t!}(x^2)^r(x)^s(2)^t,
\]
where
\[
r+s+t=8.
\]
The power of \(x\) in this term is
\[
2r+s.
\]
We find all non-negative integer solutions of
\[
2r+s=12,
\qquad
r+s+t=8.
\]
Step 1: Determine the possible values of \(r,s,t\). From \[ 2r+s=12, \] and \[ r+s+t=8, \] substituting \(s=12-2r\), \[ t=8-r-(12-2r)=r-4. \] Since \(s,t\ge 0\), \[ 12-2r\ge 0 \quad\Rightarrow\quad r\le 6, \] \[ r-4\ge 0 \quad\Rightarrow\quad r\ge 4. \] Thus, \[ r=4,5,6. \] The corresponding values are \[ (r,s,t)=(4,4,0), \] \[ (r,s,t)=(5,2,1), \] \[ (r,s,t)=(6,0,2). \]
Step 2: Find the contribution from each case. For \((4,4,0)\), \[ \frac{8!}{4!\,4!\,0!} = 70. \] For \((5,2,1)\), \[ \frac{8!}{5!\,2!\,1!}\cdot 2 = 168\cdot 2 = 336. \] For \((6,0,2)\), \[ \frac{8!}{6!\,0!\,2!}\cdot 2^2 = 28\cdot 4 = 112. \]
Step 3: Add all contributions. \[ 70+336+112 = 518. \] Therefore, the coefficient of \(x^{12}\) is \[ \boxed{518} \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Determine the possible values of \(r,s,t\). From \[ 2r+s=12, \] and \[ r+s+t=8, \] substituting \(s=12-2r\), \[ t=8-r-(12-2r)=r-4. \] Since \(s,t\ge 0\), \[ 12-2r\ge 0 \quad\Rightarrow\quad r\le 6, \] \[ r-4\ge 0 \quad\Rightarrow\quad r\ge 4. \] Thus, \[ r=4,5,6. \] The corresponding values are \[ (r,s,t)=(4,4,0), \] \[ (r,s,t)=(5,2,1), \] \[ (r,s,t)=(6,0,2). \]
Step 2: Find the contribution from each case. For \((4,4,0)\), \[ \frac{8!}{4!\,4!\,0!} = 70. \] For \((5,2,1)\), \[ \frac{8!}{5!\,2!\,1!}\cdot 2 = 168\cdot 2 = 336. \] For \((6,0,2)\), \[ \frac{8!}{6!\,0!\,2!}\cdot 2^2 = 28\cdot 4 = 112. \]
Step 3: Add all contributions. \[ 70+336+112 = 518. \] Therefore, the coefficient of \(x^{12}\) is \[ \boxed{518} \] \[ \boxed{\text{Answer = (A)}} \]
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