Let `D_(r) = |(a,2^(r),2^(16) -1),(b,3(4^(r)),2(4^(16) -1)),(c,7(8^(r)),4(8^(16) -1))|`, then the value of `underset(k =1)overset(16)Sigma D_(k)`, is (a) 0 (b) `a+b+c` (c)`ab+bc+ca` (d) 1

Options

A. 0
B. `a +b +c`
C. `ab + bc +ca`
D. none of these

Answer

A

Explanation

`underset(k=1)overset(16)Sigma D_(k) = |(a,underset(k=1)overset(16)Sigma 2^(k),2^(16) -1),(b,3 (underset(k=1)overset(16)Sigma 4^(k)),2(4^(16) -1)),(c,7 (underset(k=1)overset(16)Sigma 8^(k)),4(8^(16) -1))|`
`rArr underset(k=1)overset(16)Sigma D_(k) |(a,2((2^(16) -1)/(2 -1)),2^(16) -1),(b,3xx 4 ((4^(16) -1)/(4 -1)),2(4^(16) -1)),(c,7xx 8 ((8^(16) -1)/(8 -1)),4(8^(16) -1))|`
`rArr underset(k=1)overset(16)Sigma D_(k) =2 |(a,(2^(16) -1),(2^(16) -1)),(b,2(4^(16) 1),2(4^(16) -1)),(c,4(8^(16) -1),4(8^(16) -1))| [("Taking 2 common"),(" from " C_(2))]`
`rArr underset(k=1)overset(16)Sigma D_(k) = 2 xx 0 = 0`