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JEE Main 202227 JuneMorning Shift
Exam
JEE Main
Year
2022
Date
27 June
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2022 · 27 June · Morning Shift · Mathematics · Binomial Theorem

If the coefficient of x ^ 10 in the binomial expansion of ( x...

If the coefficient of x10{x}^{10} in the binomial expansion of (x514+5x13){\left(\dfrac{\sqrt{x}}{5\dfrac{1}{4}} + \dfrac{\sqrt{5}}{x\dfrac{1}{3}}\right)}^{60}isis{5}^{k}l,where, wherel, k \in N and ll is coprime to 55, then kk is equal to ______.

Correct answer

0

Step-by-step solution

Tr+1=Cr60(x514)60−r(5x13)r{T}_{r + 1} = Cr 60 {\left(\dfrac{\sqrt{x}}{{5}^{\dfrac{1}{4}}}\right)}^{60 - r}{\left(\dfrac{\sqrt{5}}{{x}^{\dfrac{1}{3}}}\right)}^{r}

For x10{x}^{10}, 60−r2−r3=10⇒r=24\dfrac{60 - r}{2} - \dfrac{r}{3} = 10 ⇒ r = 24

Hence coefficient of x10{x}^{10} will be C246053C24 60 {5}^{3}

Now, exponent of 55 in C2460C24 60 will be the power of 55 in 60!24!36!i.e.51454⋅58=52\dfrac{60 !}{24 ! 36 !}\text{i.e.}\dfrac{{5}^{14}}{{5}^{4} \cdot {5}^{8}} = {5}^{2}

i.e. k=5k = 5

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