Let m, n ∈ N and (2, n) = 1 . If 30( array ll 30 0 array ) + 29( array ll 30 1 array ) + . . + 2( array ll 30 28 array ) + 1( array ll 30 29 array ) = n . 2 ^ m , then n + m is equal to _______. (Here ( array ll n k

Correct answer: 0 30(C0 30) + 29(C1 30) + + 2(C28 30) + 1(C29 30) = 30(C30 30) + 29(C29 30) + + 2(C2 30) + 1(C1 30) = ∑ _ r = 1 ^ 30 r(Cr 30) = ∑ _ r = 1 ^ 30 r((30)/(r))(Cr - 1 29) = 30 ∑ _ r = 1 ^ 30 Cr - 1 29 = 30(C0 29 + C1 29 + C2 29 + + C29 29) = 30( 2 ^ 29 ) = 15 (2) ^ 30 = n (2) ^ m ∴ n = 15, m = 30 n + m = 45

JEE Main 202126 FebruaryMorning Shift

Exam: JEE Main
Year: 2021
Date: 26 February
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2021 · 26 February · Morning Shift · Mathematics · Binomial Theorem

Let m, n ∈ N and (2, n) = 1 . If 30( array ll 30 0 array ) + 29(...

Let

m, n \in N and

\gcd \left(2, n\right) = 1

. If

30\left(

\begin{array}{ll}30 \\ 0\end{array}

\right) + 29\left(

\begin{array}{ll}30 \\ 1\end{array}

\right) + \ldots . . + 2\left(

\begin{array}{ll}30 \\ 28\end{array}

\right) + 1\left(

\begin{array}{ll}30 \\ 29\end{array}

\right) = n . {2}^{m}

, then

n + m

is equal to _______. (Here \left(

\begin{array}{ll}n \\ k\end{array}

\right) = Ck n

)

Correct answer

Step-by-step solution

30\left(C0 30\right) + 29\left(C1 30\right) + \ldots + 2\left(C28 30\right) + 1\left(C29 30\right)

$= 30\left(C30 30\right) + 29\left(C29 30\right) + \ldots \ldots + 2\left(C2 30\right) + 1\left(C1 30\right)$

$= \mathop{∑}_{r = 1}^{30}r\left(Cr 30\right)$

$= \mathop{∑}_{r = 1}^{30}r\left(\dfrac{30}{r}\right)\left(Cr - 1 29\right)$

$= 30\mathop{∑}_{r = 1}^{30} Cr - 1 29$

$= 30\left(C0 29 + C1 29 + C2 29 + \ldots + C29 29\right)$

$= 30\left({2}^{29}\right) = 15{\left(2\right)}^{30} = n{\left(2\right)}^{m}$

$∴ n = 15, m = 30$

$n + m = 45$

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