Let $\binom{\mathrm{n}}{\mathrm{k}}$ denote ${}^{\mathrm{n}}\mathrm{C}_{\mathrm{k}}$ and $\left[\begin{matrix}\mathrm{n}\\\mathrm{k}\end{matrix}\right]=\begin{cases}\binom{\mathrm{n}}{\mathrm{k}} & \text{, if } 0\leq\mathrm{k}\leq\mathrm{n}\\ 0 & \text{, otherwise.}\end{cases}$ If $\mathrm{A}_{\mathrm{k}}=\sum_{i=0}^{9}\binom{9}{i}\left[\begin{matrix}12\\12-\mathrm{k}+i\end{matrix}\right]+\sum_{i=0}^{8}\binom{8}{i}\left[\begin{matrix}13\\13-\mathrm{k}+i\end{matrix}\right]$ and $\mathrm{A}_4-\mathrm{A}_3=190\ \mathrm{p}$, then p is equal to __________.

JEE Main 2021 · MathsNumerical answerNumerical valueHardMulti-step

JEE Main 26 August 2021, Shift 2, Maths Q83

Question 83 of 90 in this shift, Maths question 23 of 30, Section B.

Let

\binom{\mathrm{n}}{\mathrm{k}}

denote

{}^{\mathrm{n}}\mathrm{C}_{\mathrm{k}}

and

\left[\begin{matrix}\mathrm{n}\\\mathrm{k}\end{matrix}\right]=\begin{cases}\binom{\mathrm{n}}{\mathrm{k}} & \text{, if } 0\leq\mathrm{k}\leq\mathrm{n}\\ 0 & \text{, otherwise.}\end{cases}

\mathrm{A}_{\mathrm{k}}=\sum_{i=0}^{9}\binom{9}{i}\left[\begin{matrix}12\\12-\mathrm{k}+i\end{matrix}\right]+\sum_{i=0}^{8}\binom{8}{i}\left[\begin{matrix}13\\13-\mathrm{k}+i\end{matrix}\right]

and

\mathrm{A}_4-\mathrm{A}_3=190\ \mathrm{p}

, then p is equal to __________.

Official answer

NTA final key.

Chapter: Binomial Theorem
Topic: Binomial Coefficients
Idea tested: Vandermonde's Identity

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Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.