So, statement-1 is also true. Stetement-2 is a correct
expanation for statement-1.
`S_(1)= sum_(j=1)^(10) j (j -1)""^(10)C_(j),S_(2)= sum_(j=1)^(10)j.""^(10)C_(j) and S_(2)= sum_(j=1)^(10)j.""^(10)C_(j) .` Statement-1 `S_(3) = 50xx2^(9)`.
Statement-2 `S_(1) = 90xx2^(8) and S_(2) = 10 xx 2^(8)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

d

Explanation

We have, `S_(2)= sum_(j=1)^(10) j ""^(10)C_(j)=sum_(j=1)^(10) jxx(10)/(j) xx(10)/(j) ""^(9)C_(j-1) = 10 sum_(j=1)^(10)""^(9)C_(j-1) = 10 xx2^(9)`
`S_(1)= sum_(j=1)^(10) (j-1) ""^(10)C_(j)=sum_(j=1)^(10) j(j-1)xx(10)/(j) xx(9)/(j-1) xx""^(8)C_(j-2)`
`rArr S_(2) = 10 xx9 sum_(j=2)^(10) ""^(8)C_(j-2) = 90 xx2^(8)`
and, `s_(3) = sum_(j=1)^(10) j^(2) ""^(10)C_(j) = sum_(j=1)^(10){j (j -1)+ j} ""^(10)C_(j)`
`rArr S_(3) = sum_(j = 1)^(10) j(j -1)""^(10)C_(j) + sum_(j=1)^(10) j ""^(10)C_(j) = 10 xx9 2^(8) + 10 xx2^(9)`
`rArr S_(3) = 11 xx10 xx2^(8)`
So, statement-2 is true and statement-1 is false.