show that the number of ways one can choose a set of distinct positive integers, each smaller than or equal to 50 ,such that their sum is odd,is 2^49

Dear pritam

we have 25 even number and 25 odd number

for sum to be odd ,we must select odd number of odd number .

so numere of ways =

                       ²⁵C₀(²⁵C₁ +²⁵C₃+²⁵C₅+²⁵C_{7 .........................}+²⁵C₂₅₎

                  +  ²⁵C₁(²⁵C₁ +²⁵C₃+²⁵C₅+²⁵C_{7 .........................}+²⁵C₂₅₎

                  +  ²⁵C₂(²⁵C₁ +²⁵C₃+²⁵C₅+²⁵C_{7 .........................}+²⁵C₂₅₎

                  .

                  .

                  .

                 .

                    +  ²⁵C₂₅(²⁵C₁ +²⁵C₃+²⁵C₅+²⁵C_{7 .........................}+²⁵C₂₅₎

          =  (²⁵C₀ +   ²⁵C₁ +²⁵C₂+²⁵C₃+²⁵C_{4 .........................}+²⁵C₂₅₎(²⁵C₁ +²⁵C₃+²⁵C₅+²⁵C_{7 ........................}+²⁵C₂₅₎

          =2²⁵ .2^25-1

          =2⁴⁹

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