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JEE Main 202124 FebruaryMorning Shift
Exam
JEE Main
Year
2021
Date
24 February
Shift
Morning Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2021 · 24 February · Morning Shift · Mathematics · Binomial Theorem

The value of - C1 15 + 2 · C2 15 - 3 · ^ 15 C _ 3 + . . . . . - 15...

The value of −C115+2⋅C215−3⋅15C3+.....−15⋅C1515+C114+C314+C514+....+C1114- C1 15 + 2 \cdot C2 15 - 3{\cdot}^{15}{C}_{3} + . . . . . - 15 \cdot C15 15 + C1 14 + C3 14 + C5 14 + . . . . + C11 14 is equal to

Options

  1. A214{2}^{14}
  2. B213−13{2}^{13} - 13
  3. C216−1{2}^{16} - 1
  4. D213−14{2}^{13} - 14

Correct answer

D. 213−14{2}^{13} - 14

Step-by-step solution

(−C115+2⋅C215−3⋅C315+…−15⋅C1515)+\left(- C1 15 + 2 \cdot C2 15 - 3 \cdot C3 15 + \ldots - 15 \cdot C15 15\right) +\left(C1 14 + C3 14 + \ldots + C11 14\right)

=\left(\mathop{∑}_{r = 1}^{15}{(−1)\left(- 1\right)}^{r} \cdot r{\cdot}^{15}{C}_{r}\right) + \left(C1 14 + C3 14 + \ldots + C11 14 + C13 14\right)−C1314- C13 14

=(∑r=115(−1)r⋅15⋅Cr−114)+213−14= \left(\mathop{∑}_{r = 1}^{15}{\left(- 1\right)}^{r} \cdot 15 \cdot Cr - 1 14\right) + {2}^{13} - 14

=15(−C014+C114……..−C1414)+213−14= 15\left(- C0 14 + C1 14 \ldots \ldots . . - C14 14\right) + {2}^{13} - 14

=213−14= {2}^{13} - 14

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