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04/01/2022 12:43 pm
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Find: \(\sum_{n = 0}^{20}(^{20}C_n)^2\)
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04/01/2022 12:54 pm
\(\sum_{n=0}^{20}(^{20}C_n)^2\)
\((^{20}C_n)^2\)→ coefficient of xr in (1+x)20 expansion.
(1+x)20 = \(^{20}C_0\) + \(^{20}C_1x + {20}C_2x^2 + \).....+ \(^{20}C_{20}x^{20}\)
(1+x)20 = \(^{20}C_0x^{20}\) + \(^{20}C_1x^{19} + ^{20}C_2x^{18} + \).....+ \(^{20}C_{20}\)
Multiplying both the equations,
(1+x)40 = \(^{20}C_0\) + \(^{20}C_1x + ^{20}C_2x^2 + \).....+ \(^{20}C_20x^{20}\)x\(^{20}C_0\) + \(^{20}C_1x^{19} + ^{20}C_2x^{18} + \).....+ \(^{20}C_{20}\)
Comparing both the coefficients of x20 on both the sides,
\(^{40}C_{20}\) = \((^{20}C_0)^2\) + \((^{20}C_1)^2\) + .......+ \(^{20}C_{20}\)
\(^{40}C_{20}\) = \(\sum_{n=0}^{20}(^{20}C_n)^2\)