Sum of the last 12 coefficients in the binomial expansion of (1+x)23 is

Sum of last 12 coefficients of (1+x)23 is S=23C12+23C13+……+23C23…..(1) Using nCr=nCn−r we get S=23C11+23C10+…..+23C0……(2)Adding (1) and (2),  we get 2S=23C0+23C1+…..23C23=223⇒S=222∴ Sum of last 12 coefficients of (1+x)23 is 222