Properties of Binomial coefficients
Binomial Theorem: General form of a polynomial in x is an xⁿ + an₋₁xⁿ⁻¹ + … + a₁x + a₀, where an, an₋₁, …, a₁, a₀, are constants, an ≠ 0 and n is a whole number. A binomial is a polynomial which consists of two terms.
Use Pascal’s triangle
Properties of Binomial Coefficients
(i) If ⁿCr = ⁿCs then either r = s (or) r + s = n
(ii) ⁿC₀ + ⁿC₁ + … + ⁿCn = 2ⁿ
(iii) ⁿC₀ + ⁿC₂ + ⁿC₄ + … = ⁿC₁ + ⁿC₃ + … = 2ⁿ⁻¹
(iv) ⁿC₀ – ⁿC₁ + ⁿC₂ – ⁿC₃ + … (-1)ⁿ ⁿCn = 0
(v) ⁿCr + ⁿCr₋₁ = ⁿ⁺¹Cr
(v) \(\frac{^{n}{{C}_{r}}}{^{n}{{C}_{r-1}}}=\frac{n-r+1}{r}\)
(vi) ⁿ⁺¹Cr₊₁ = \(\frac{n+1}{r+1}{{\,}^{n}}{{C}_{r}}\)
(vii) ⁿC₁ + 2 ⁿC₂ + 3 ⁿC₃ + … + n ⁿCn = n.2ⁿ⁻¹
(viii) ⁿC₁ – 2 ⁿC₂ + 3 ⁿC₃ – … = 0
(ix) ⁿC₀ + 2 ⁿC₁ + 3 ⁿC₂ + … + (n + 1) ⁿCn = (n + 2) 2ⁿ⁻¹
(x) C₀Cr + C₁Cr₊₁ + … + Cn₋₁ Cn = \(\frac{\left( 2n \right)!}{\left( n-r \right)!\left( n+r \right)!}\)
(xi) C²₀ + C²₁ + C²₂ + … + C²n \(\frac{\left( 2n \right)!}{{{\left( n! \right)}^{2}}}\)
(xii) C²₀ – C²₁ + C²₂ – C²₃ + … = \(\left\{ \begin{align}& 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,n\,is\,odd \\& {{\left( -1 \right)}^{n/2}},\,\,\,if\,n\,is\,even \\\end{align} \right.\).
Example: \(~\sum\limits_{r=0}^{n}{{{3}^{r}}}\)ⁿCr is equal to
Solution: \(\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}}\)x 3r
= ⁿC₀ 3⁰ + ⁿC₁ 3¹ + ⁿC₂ 3² + ⁿC₃ 3³ + ⁿCn 3ⁿ
= (1 + 3)ⁿ
= 4ⁿ
Example: The value of \(^{47}{{C}_{4}}+\sum\limits_{i=0}^{3}{^{50-j}{{C}_{3}}}+\sum\limits_{k=0}^{5}{^{56-k}{{C}_{53-k}}}\).
Solution: \(^{47}{{C}_{4}}+\sum\limits_{i=0}^{3}{^{50-j}{{C}_{3}}}+\sum\limits_{k=0}^{5}{^{56-k}{{C}_{53-k}}}\).
= ⁴⁷C₄ + (⁵⁰C₃ + ⁴⁹C₃ + ⁴⁸C₃ + ⁴⁷C₃) + (⁵⁶C₅₃ + ⁵⁵C₅₂ + ⁵⁴C₅₁ + ⁵³C₅₀ + ⁵²C₄₉ + ⁵¹C₄₈)
= ⁴⁷C₄ + (⁴⁷C₃ + ⁴⁸C₃ + ⁴⁹C₃ + ⁵⁰C₃) + (⁵⁶C₃ + ⁵⁵C₃ + ⁵⁴C₃ + ⁵³C₃ + ⁵²C₃ + ⁵¹C₃) [∵ ⁿCr = ⁿCn-r]
= (⁴⁷C₄ + ⁴⁷C₃) + (⁴⁸C₃ + ⁴⁹C₃ + ⁵⁰C₃) + (⁵¹C₃ + ⁵²C₃ + … + ⁵⁶C₃) [∵ ⁿCr + ⁿCr₋₁ = ⁿ⁺¹Cr]
= (⁴⁸C₄ + ⁴⁸C₃) + ⁴⁹C₃ + ⁵⁰C₃ +… + ⁵⁶C₃ = ⁵⁷C₄