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Multinomial Expansion

Q. What is th...

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Q. What is the sum of the binomial coefficients in the expansion of ${(1 + x)}^{50}$ ?
Q. What is the sum of the coefficients of odd powers of x in the expansion of ${(1 + x)}^{50}$ ?

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Solution

$(1) . {(1 + x)}^{50} = {}^{50}C_{0} . 1^{50} . x^{0} + {}^{50}C_{1} . 1^{49} . x^{1} + {}^{50}C_{2} . 1^{48} . x^{2} + . . . . . . . + {}^{50}C_{50} . 1^{0} . x^{50} Putting x = 1, we get {(1 + 1)}^{50} = {}^{50}C_{0} + {}^{50}C_{1} + {}^{50}C_{2} + . . . . . . . + {}^{50}C_{50} \Rightarrow {}^{50}C_{0} + {}^{50}C_{1} + {}^{50}C_{2} + . . . . . . . + {}^{50}C_{50} = 2^{50} So, sum of binomial coefficients in the expansion of {(1 + x)}^{50} is 2^{50} . (2) . {(1 + x)}^{50} = {}^{50}C_{0} . 1^{50} . x^{0} + {}^{50}C_{1} . 1^{49} . x^{1} + {}^{50}C_{2} . 1^{48} . x^{2} + . . . . . . . + {}^{50}C_{50} . 1^{0} . x^{50} Putting x = - 1, we get {(1 - 1)}^{50} = {}^{50}C_{0} - {}^{50}C_{1} + {}^{50}C_{2} - {}^{50}C_{3} + . . . . . . . + {}^{50}C_{50} \Rightarrow 0 = ({}^{50}C_{0} + {}^{50}C_{2} + {}^{50}C_{4} + . . . . . .) - ({}^{50}C_{1} + {}^{50}C_{3} + {}^{50}C_{5} + . . . . . .) \Rightarrow {}^{50}C_{0} + {}^{50}C_{2} + {}^{50}C_{4} + . . . . . . + {}^{50}C_{50} = {}^{50}C_{1} + {}^{50}C_{3} + {}^{50}C_{5} + . . . . . . . . + {}^{50}C_{49} . . . . . . . (1) \Rightarrow sum of coefficients of even terms in the expansion of {(1 + x)}^{50} = sum of coefficients of odd terms in the expansion of {(1 + x)}^{50} \Rightarrow {}^{50}C_{0} + {}^{50}C_{1} + {}^{50}C_{2} + . . . . . . . + {}^{50}C_{50} = 2 ({}^{50}C_{1} + {}^{50}C_{3} + {}^{50}C_{5} + . . . . . . . . + {}^{50}C_{49}) \Rightarrow {}^{50}C_{1} + {}^{50}C_{3} + {}^{50}C_{5} + . . . . . . . . + {}^{50}C_{49} = \frac{2^{50}}{2} = 2^{49} \Rightarrow sum of coefficients of odd terms in the expansion of {(1 + x)}^{50} is 2^{49} .$

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