The value of $(^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + . . . + (^{21} C_{10} -^{10} C_{10})$ is

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Solution

$\begin{matrix} (^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + . . . + (^{21} C_{10} -^{10} C_{10}) = (^{21} C_{1} +^{21} C_{2} +^{21} C_{3} + {. . . . .}^{21} C_{10})      I - (^{10} C_{1} +^{10} C_{2} +^{10} C_{3} + . . . +^{10} C_{10})      2^{10} - 1 I =^{21} C_{1} +^{21} C_{2} +^{21} C_{3} + . . . . .^{21} C_{10} I =^{21} C_{20} +^{21} C_{19} +^{21} C_{18} + . . . . .^{21} C_{20} 2 I =^{21} C_{1} +^{21} C_{2} +^{21} C_{3} + . . . . . +^{21} C_{19} +^{21} C_{20} 2 I = 2^{21} - 1 I = \frac{2^{21} - 1}{2} = 2^{20} - \frac{1}{2} ∴ I - 2^{10} - 1 = 2^{20} - \frac{1}{2} - 2^{10} - 1 = 2^{20} - 2^{10} - \frac{3}{2} \end{matrix}$

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