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JEE Main 202228 JulyEvening Shift
Exam
JEE Main
Year
2022
Date
28 July
Shift
Evening Shift
Subject
Mathematics
Chapter
Binomial Theorem

JEE Main 2022 · 28 July · Evening Shift · Mathematics · Binomial Theorem

If 1 + (2 + C1 49 + C2 49 + . + C49 49) (C2 50 + C4 50 + . . + C50...

If 1+1 +\left(2 + C1 49 + C2 49 + \ldots . + C49 49\right)(C250+C450+…..+C5050)\left(C2 50 + C4 50 + \ldots . . + C50 50\right) is equal to 2n.m{2}^{n} . m, where mm is odd, then n+mn + m is equal to _____ .

Correct answer

0

Step-by-step solution

Let I=1+(1+C049+C149+…+C4949)(C250+C450+…+C5050)I = 1 + \left(1 + C0 49 + C1 49 + \ldots + C49 49\right)\left(C2 50 + C4 50 + \ldots + C50 50\right)

As C049+C149+…+C4949=249C0 49 + C1 49 + \ldots + C49 49 = {2}^{49}

and C050+C250+…+C5050=249C0 50 + C2 50 + \ldots + C50 50 = {2}^{49}

⇒C250+C450+….+C5050=249−1⇒ C2 50 + C4 50 + \ldots . + C50 50 = {2}^{49} - 1

∴I=1+(249+1)(249−1)∴ I = 1 + \left({2}^{49} + 1\right)\left({2}^{49} - 1\right)= {2}^{98}</p><p>Nowcomparing</p><p>Now comparingI = {2}^{n}mweget,we get,m = 1andandn = 98</p><p>So,</p><p>So,m + n = 99

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