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The sum sum(k=1)^(10)underset(i ne j ne ...

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

A

`240`

B

`720`

C

`540`

D

`1080`

Text Solution

Verified by Experts

The correct Answer is:
B
`(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`
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