Solving JEE Advanced Problems in Python: Exploring Binomial Expansion
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Today, we will examine a problem that appeared in JEE Advanced 2016, specifically in Paper I as problem number 51 in the Part III: Mathematics section. You can find the official question paper linked here.
Let \( m \) be the smallest positive integer such that the coefficient of \( x^2 \) in the expansion of $$ (1+x)^{2}+(1+x)^{3}+\cdots+(1+x)^{49}+(1+mx)^{50} $$ is \( (3n+1)^{51}C_{3} \) for some positive integer n. Then the value of \( n \) is
The binomial theorem is a mathematical formula that expresses the expansion of a power of the sum of two numbers. It states that the expansion of \( (a+b)^n \) can be expressed as a sum of terms, each of which is the product of a binomial coefficient and powers of \( a \) and \( b \).
The binomial coefficient, denoted by \( {n \choose k} \), is the number of ways to choose \( k \) objects from a set of \( n \) objects. The formula for the binomial theorem is:
