Let `S_(1) = sum__(j=1)^(10) j(j-1)""^(10)C_(j), S_(2) = sum_(j=1)^(10)""^(10)C_(j)`, and `S_(3) = sum_(j=1)^(10) j^(2).""^(10)C_(j)`.
Statement 1 : `S_(3) xx 2^(9)`.
Statement 2 : `S_(1) = 90 xx 2^(8)` and `S_(2) = 10 xx 2^(8)`.

Options

A. Statement 1 is false, statement 2 is true.
B. Statement 1 is true, statement 2 is true, statement 2 is a correct explanation for statement 1.
C. Statement 1 is true, statement 2 is true, statement 2 is not a correct explanation for statement 2.
D. statement 1 is true, statement 2 is false.

Answer

B

Explanation

`S_(1)=underset(j=1)overset(10)sumj(j-1)(10!)/(j(j-1)(j-2)!(10-j)!)`
`= 90underset(j=1)overset(10)sum(8!)/((j-2)!(8-(j-2)!)`
`=90underset(j=2)overset(10)sum.^(8)C_(j-2)=90xx2^(8)`
`S_(1) = underset(j=1)overset(10)sum(10!)/(j(j-1)!(9-(j-1))!)`
`= 10underset(j=1)overset(10)sum(9!)/((j-1)!(9-(j-1))!)`
`10underset( j=1)overset(10)sum.^(9)C_(j-1)= 10 xx 2^(9)`
`S_(3) = underset(j=1)overset(10)sum[j(j-1)+j] .^(10)C_(j)`
`= underset(j=1)overset(10)sumj(j-1).^(10)C_(j)+underset(j=1)overset(10)sum..^(10)C_(j)`
`= 90xx2^(8)+10xx2^(9) = 55 xx 2^(9)`