The coefficient of x 5 in the expansion of \((1+\text{x})^{21}+(1+\text{x})^{22}+...+(1+\text{x})^{30}\)

The coefficient of x5 in the expansion of \((1+\text{x})^{21}+(1+\text{x})^{22}+...+(1+\text{x})^{30}\)

  1. \({^\text{9}}\text{C}_{\text{5}}\)

  2. \({^\text{51}}\text{C}_{\text{5}}\)

  3. \({^\text{31}}\text{C}_{\text{6}}-{^\text{21}}\text{C}_{\text{6}}\)

  4. \({^\text{30}}\text{C}_{\text{5}}+{^\text{20}}\text{C}_{\text{5}}\)

Solution

Solution:

we have,

\((1+\text{x})^{21}+(1+\text{x})^{22}+...+(1+\text{x})^{30}\)

\(=(1+\text{x})^{21}\Big[\frac{(1+\text{x})^{10}-1}{(1+\text{x})+1}\Big]\)

\(=\frac{1}{\text{x}}\Big[(1+\text{x})^{31}-(1+\text{x})^{21}\Big]\)

Coefficient of x5 in the given expansion = Coefficient of x5 in \(=\frac{1}{\text{x}}\Big[(1+\text{x})^{31}-(1+\text{x})^{21}\Big]\)

= Coefficient of x6 in \(\Big[(1+\text{x})^{31}-(1+\text{x})^{21}\Big]\)

\(={^\text{31}}\text{C}_{\text{6}}-{^\text{21}}\text{C}_{\text{6}}\)

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