The coefficient of x^ 70 in x^2(1+x)^ 98 +x^3(1+x)^ 97 +x^4(1+x)^ 96 + +x^ 54 (1+x)^ 46 is ^ 99 C_ p - ^ 46 C_ q . Then a possible value of p+q is :

Correct answer: B. 83 aligned & x^2(1+x)^ 98 +x^3(1+x^ 97 )+x^4(1+x)^ 96 + & x^ 54 (1+x)^ 46 aligned Coeff. of x^ 70 : ^ 98 C_ 68 + ^ 97 C_ 67 + ^ 96 C_ 66 + aligned & ^ 47 C_ 17 + ^ 46 C_ 16 & = ^ 46 C_ 30 + ^ 47 C_ 30 + . . ^ 98 C_ 30 & =( ^ 46 C_ 31 + ^ 46 C_ 30 )+ ^ 47 C_ 30 + . ^ 98 C_ 30 - ^ 46 C_ 31 & = ^ 47 C_ 31 + ^ 47 C_ 30 + . . ^ 98 C_ 30 - ^ 46 C_ 31 & & = ^ 99 C_ 31 - ^ 46 C_ 31 = ^ 99 C_ p - ^ 46 C_ q aligned Possible values of (p+q) are 62,83,99,46 p+q=83

JEE Main 20249 AprilMorning Shift

Exam: JEE Main
Year: 2024
Date: 9 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Binomial Theorem

The coefficient of x^ 70 in x^2(1+x)^ 98 +x^3(1+x)^ 97 +x^4(1+x)^...

The coefficient of

x^{70}

x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}

{ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}

. Then a possible value of

p+q

is :

Options

Correct answer

B. 83

Step-by-step solution

\begin{aligned} & x^2(1+x)^{98}+x^3\left(1+x^{97}\right)+x^4(1+x)^{96}+\ldots \ldots \\ & x^{54}(1+x)^{46} \end{aligned}

Coeff. of

\mathrm{x}^{70}:{ }^{98} \mathrm{C}_{68}+{ }^{97} \mathrm{C}_{67}+{ }^{96} \mathrm{C}_{66}+\ldots \ldots \ldots

\begin{aligned} & { }^{47} \mathrm{C}_{17}+{ }^{46} \mathrm{C}_{16} \\ & ={ }^{46} \mathrm{C}_{30}+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}_{30} \\ & =\left({ }^{46} \mathrm{C}_{31}+{ }^{46} \mathrm{C}_{30}\right)+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots .{ }^{98} \mathrm{C}_{30}-{ }^{46} \mathrm{C}_{31} \\ & ={ }^{47} \mathrm{C}_{31}+{ }^{47} \mathrm{C}_{30}+\ldots \ldots \ldots . .{ }^{98} \mathrm{C}_{30}-{ }^{46} \mathrm{C}_{31} \\ & \ldots \ldots \\ & ={ }^{99} \mathrm{C}_{31}-{ }^{46} \mathrm{C}_{31}={ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}} \end{aligned}

<br />Possible values of

(p+q)

are

62,83,99,46 \Rightarrow p+q=83$

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