The sum `sum_(k=1)^(10)underset(i ne j ne k)underset(j=1)(sum^(10))sum_(i=1)^(10)1` is equal to

Options

A. `240`
B. `720`
C. `540`
D. `1080`

Answer

B

Explanation

`(b)` `sum_(k=1)^(10)underset(i nej ne k)(sum_(j=1)^(10))sum_(i=1)^(10)1`
`=`(sum when `i`, `j`, `k` are independent)
`-`(sum whenany two of `i`,`j`,`k` are equal)
`-`(sum when all `i`, `j`, `k` are equal)
`=sum_(k=1)^(10)underset(i nej ne k)(sum_(j=1)^(10))sum_(i=1)^(10)1-3sum_(k=1)^(10)underset(i=j)(sum_(i=1)^(10))1+2underset(i=j=k)(sum_(k=1)^(10))1`
`=10^(3)-3xx10^(2)+2xx10`
`=720`