How many coefficients are odd in the expansion of ?
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By binomial expansion, ( x + 1 ) 1 0 0 0 = ( 0 1 0 0 0 ) x 1 0 0 0 + ( 1 1 0 0 0 ) x 9 9 9 + ( 2 1 0 0 0 ) x 9 9 8 + ⋯ + ( 1 0 0 0 1 0 0 0 ) x 0 .
So we want to find the number of odd numbers among the set of numbers, { ( 0 1 0 0 0 ) , ( 1 1 0 0 0 ) , ( 2 1 0 0 0 ) , … , ( 1 0 0 0 1 0 0 0 ) } .
Referencing Lucas' theorem :
If we rewrite 1000 in base 2, we have 1 0 0 0 1 0 = 1 1 1 1 1 0 1 0 0 0 2 . Since there's six 1's in its binary representation, the answer is 2 6 = 6 4 .