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 𝗕𝗶𝗻𝗼𝗺𝗶𝗮𝗹 𝗧𝗵𝗲𝗼𝗿𝗲𝗺 1. Introduction to Binomial Theorem The binomial theorem is a fundamental concept in algebra, which allows us to expand expressions of the form (a + b)^n, where "a" and "b" are constants and "n" is a positive integer. 2. Binomial Theorem for Positive Integral Exponents Binomial Theorem Formula: For any positive integer "n," the expansion of (a + b)^n is given by: (a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n) * a^0 * b^n Where: "C(n, k)" represents the binomial coefficient, calculated as C(n, k) = n! / (k!(n-k)!). "a^n" represents "a" raised to the power of "n." "b^k" represents "b" raised to the power of "k." 3. Middle Term(s) in Binomial Expansion To find the middle term(s) in the binomial expansion, we use the formula for the middle term: Middle term(s) = (n + 1) / 2 if "n" is odd...

𝗨𝗻𝗱𝗲𝗿𝘀𝘁𝗮𝗻𝗱𝗶𝗻𝗴 𝗔𝘁𝗼𝗺𝗶𝗰 𝗦𝘁𝗿𝘂𝗰𝘁𝘂𝗿𝗲 The Atom: Nature's Fundamental Unit The atom is the smallest unit of an element that retains its chemical properties. This concept is at the heart of chemistry and has a rich history of scientific discovery. Let's dive into the key aspects of atomic structure: 1. Historical Background The journey of understanding the atom began with ancient Greek philosophers who pondered the idea of indivisible particles. John Dalton formulated the first comprehensive atomic theory. His postulates proposed that atoms are indivisible, and chemical reactions involve the rearrangement of atoms, giving birth to modern chemistry. 2. Subatomic Particles ● Electrons: J.J. Thomson's cathode ray experiment revealed the existence of negatively charged electrons within atoms. ● Protons: Ernest Rutherford's gold foil experiment showed that atoms also contain positively charged protons concentrated in a tiny nucleus at the center.  ● Neu...