Seven people leave their bags outside al temple and returning after worshiping picked one bag each at random. In how many ways at least one and at most three of them get their correct bags?

Explanation

Number of ways in which only one person get his own
`=""^(7)C_(1)xx("Derangement of " 6 " objects")`
`= ""^(7)C_(1)xx6!(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!)+(1)/(6!))`
`=7xx265=1855`
Number of ways in which exactly two persons get their own bag
`= ""^(7)C_(2)xx("Derangement of " 5 " objects")`
`=""^(7)C_(2)xx5!(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!)-(1)/(5!))`
`=21xx44`
=924
Number of ways in which exactly three persons get their own bag
`=""^(7)C_(3)xx("Derangement of " 4 " objects")`
`= ""^(7)C_(3)xx4!(1-(1)/(1!)+(1)/(2!)-(1)/(3!)+(1)/(4!))`
`=35xx9`
=315
`therefore` Required number of ways =1855+924+315
=3094