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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

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

Grade:12

1 Answers

Badiuddin askIITians.ismu Expert
148 Points
12 years ago

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+25C7 .........................+25C25)

                  +  25C1(25C1 +25C3+25C5+25C7 .........................+25C25)

                  +  25C2(25C1 +25C3+25C5+25C7 .........................+25C25)

                  .

                  .

                  .

                 .

                    +  25C25(25C1 +25C3+25C5+25C7 .........................+25C25)

          =  (25C0 +   25C1 +25C2+25C3+25C4 .........................+25C25)(25C1 +25C3+25C5+25C7 ........................+25C25)

          =225 .225-1

          =249

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Badiuddin

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