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High School
By Shiva Kintali on Sept. 27, 2013 | Updated Jan. 4, 2018

Binomial coefficients

Evaluate the following sums using combinatorial methods and algebraic methods : \(\displaystyle \sum_{i=0}^{k} {m \choose i}{n \choose k-i}\) \(\displaystyle \sum_{i=0}^{n} {n \choose i}^2\) …
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High School
By True IITJEE on Jan. 17, 2017 | Updated Jan. 3, 2018

JEE Advanced 2016 Paper 1 Mathematics Question 51

Let \(m\) be the smallest positive integer such that the coefficient of \(x^2\) in the expansion of \((1+x)^2 + (1+x)^3 + \dots + (1+x)^{49} + (1+mx)^{50}\) is \((3n+1)\ \ {}^{51}C_{3}\) for some posi…
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High School
By Shiva Kintali on Sept. 30, 2013 | Updated Jan. 4, 2018

Using Binomial Theorem

Prove the following using binomial theorem and/or mathematical induction Let \(a\), \(b\) and \(n\) be natural numbers, prove that \(\frac{(a+\sqrt{b})^n + (a-\sqrt{b})^n}{2}\) is also a natural …
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