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Binomial Theorem question

2022 · 25 Jun · Shift 2 · Q29
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  5. /2022 · 25 Jun · Shift 2 · Q29

Binomial Theorem question

2022 · 25 Jun · Shift 2 · Q29

JEE MainMathematicsBinomial TheoremMCQ+4 / −1
The coefficient of x101 in the expression (5+x)500+x(5+x)499+x2(5+x)498+  .....  +  x500{(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}(5+x)500+x(5+x)499+x2(5+x)498+.....+x500, x > 0, is
  1. A
    501C101 (5)399
  2. B
    501C101 (5)400
  3. C
    501C100 (5)400
  4. D
    500C101 (5)399
View written solutionFree

Correct answer: A

  1. Write the given expression in summation form

The expression is

(5+x)500+x(5+x)499+x2(5+x)498+⋯+x500.(5+x)^{500}+x(5+x)^{499}+x^2(5+x)^{498}+ \cdots +x^{500}.(5+x)500+x(5+x)499+x2(5+x)498+⋯+x500.

This can be written as

∑k=0500xk(5+x)500−k.\sum_{k=0}^{500} x^k(5+x)^{500-k}.k=0∑500​xk(5+x)500−k.
  1. Recognize it as a geometric-type factorization

Let a=5+x,b=x.a=5+x, \qquad b=x.a=5+x,b=x. Then the sum becomes

a500+ba499+b2a498+⋯+b500.a^{500}+ba^{499}+b^2a^{498}+\cdots+b^{500}.a500+ba499+b2a498+⋯+b500.

Using

an+an−1b+an−2b2+⋯+bn=an+1−bn+1a−b,a^n+a^{n-1}b+a^{n-2}b^2+\cdots+b^n=\frac{a^{n+1}-b^{n+1}}{a-b},an+an−1b+an−2b2+⋯+bn=a−ban+1−bn+1​,

we get

∑k=0500xk(5+x)500−k=(5+x)501−x501(5+x)−x.\sum_{k=0}^{500} x^k(5+x)^{500-k} =\frac{(5+x)^{501}-x^{501}}{(5+x)-x}.k=0∑500​xk(5+x)500−k=(5+x)−x(5+x)501−x501​.

Since (5+x)−x=5(5+x)-x=5(5+x)−x=5,

S=(5+x)501−x5015.S=\frac{(5+x)^{501}-x^{501}}{5}.S=5(5+x)501−x501​.
  1. Find the coefficient of x101x^{101}x101

Now x501x^{501}x501 does not contribute to the coefficient of x101x^{101}x101. So we only need the coefficient of x101x^{101}x101 in

(5+x)5015.\frac{(5+x)^{501}}{5}.5(5+x)501​.

Coefficient of x101x^{101}x101 in (5+x)501(5+x)^{501}(5+x)501 is

\binom{501}{101}5^{501-101}=inom{501}{101}5^{400}.

Therefore, after dividing by 555, the required coefficient is

\frac{1}{5}\binom{501}{101}5^{400}=inom{501}{101}5^{399}.
  1. Match with options

This is exactly:

(501101)5399\boxed{\binom{501}{101}5^{399}}(101501​)5399​

which corresponds to Option A.

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