Choose the correct option regarding the following statements 1. (C₀+C₂+C₄+ +C_n=2^ n-1 ), if (n ) is even 2. (C₁+C₃+C₅+ +C_ n-1 =2^ n-1 ), if (n ) is even

Step-by-step solution

Since, ((1+x)^n=C₀+C₁ x+C₂ x^2+C₃ x^3+ +C_n x^n ) On putting (x=1 ) and (x=-1 ) respectively, we get ( aligned 2^n & =C₀+C₁+C₂+C₃+ +C_n 0 & =C₀-C₁+C₂-C₃+ +(-1)^n C_n aligned ) and (0=C₀-C₁+C₂-C₃+ +(-1)^n C_n ) If (n ) is even , then (C₀+C₁+C₂+C₃+ +C_n=2^n )...(i) ( and C₀-C₁+C₂-C₃+ +C_n=0 )...(ii) By adding and subtracting Eqs. (i) and (ii), we get (C₀+C₂+C₄+ +C_n=2^ n-1 ) and ( C₁+C₃+C₅+ +C_ n-1 =2^ n-1 )