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Questions  >>  CBSE XI  >>  Math  >>  Binomial Theorem
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Q)

Middle term in the expansion of $(a^3+ba)^{28}$ is ______

$\begin{array}{1 1}(A)\;28C_{14}a^{56}b^{14}\\(B)\;28C_{14}\\(C)\;28C_{14}a^{56}\\(D)\;28C_{14} b^{14}\end{array} $

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A)
Toolbox:
  • If n is even then the total number of terms in the expansion of $(a+b)^n$ is n+1
  • Hence there is only one middle term (i.e) $(\large\frac{n}{2}$$+1)^{th}$.
  • $T_{r+1}=nC_r a^{n-r} b^r$
Since the power of binomial is even it has one middle term which is the $(\large\frac{n}{2}$$+1) ^{th}\Rightarrow (\large\frac{28}{2}$$+1)^{th}$
$\Rightarrow 15^{th}$
$T_{r+1}=nC_r a^{n-r} b^r$
$T_{15}=28C_{14}(a^3)^{28-14}(ba)^{14}$
$\Rightarrow 28C_{14} (a^3)^{14} (ba)^{14}$
$\Rightarrow 28C_{14} a^{42} b^{14}a^{14}$
$\Rightarrow 28C_{14} a^{56} b^{14}$
Hence (A) is the correct answer.
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