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JEE Main 20203 SeptemberMorning Shift
Exam
JEE Main
Year
2020
Date
3 September
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2020 · 3 September · Morning Shift · Mathematics · Binomial Theorem

The value of ( 2 ⋅ ^ 1 P _ 0 - 3 ⋅ ^ 2 P _ 1 + 4 ⋅ ^ 3 P _ 2 - . ....

The value of (2⋅1P0−3⋅2P1+4⋅3P2−........2{⋅}^{1}{P}_{0} - 3{⋅}^{2}{P}_{1} + 4{⋅}^{3}{P}_{2} - . . . . . . . . up to 51th{51}^{th} term)++( 1!−2!+3!−.......1 ! - 2 ! + 3 ! - . . . . . . . up to 51th{51}^{th} term) is equal to

Options

  1. A1−51(51)!1 - 51\left(51\right) !
  2. B1+(51)!1 + \left(51\right) !
  3. C1+(52)!1 + \left(52\right) !
  4. D11

Correct answer

C. 1+(52)!1 + \left(52\right) !

Step-by-step solution

2⋅P01=2×1!1!=2!2 \cdot P0 1 = 2 \times \dfrac{1 !}{1 !} = 2 !

3⋅P12=3×2!1!=3!3 \cdot P1 2 = 3 \times \dfrac{2 !}{1 !} = 3 !

4⋅P23=4×3!1!=4!4 \cdot P2 3 = 4 \times \dfrac{3 !}{1 !} = 4 !

By this observation given question can be written as

=(2!−3!+4!−5!+.......−51!+52!)+(1!−2!+3!−4!+......−50!+51!)= \left(2 ! - 3 ! + 4 ! - 5 ! + . . . . . . . - 51 ! + 52 !\right) + \left(1 ! - 2 ! + 3 ! - 4 ! + . . . . . . - 50 ! + 51 !\right)

=1!+52!=1+52!= 1 ! + 52 ! = 1 + 52 !

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