An analysis of 100 personal injury claims made upon a motor insurance company revealed that loss or injury in respect of an eye, an arm, a leg occurred in 30, 50 and 70 cases, respectively. Claims involving this loss or injury to two of these members numbered 44. How many claims involved loss or injury to all the three, we must assume that one or another of three members was mentioned in each of the 100 claims?

Explanation

Let the set of people having injuries in eyes, arms or legs be denoted by E, A and L, respectively. Then, according to the problem, we have
`n(EuuAuuL)=30,n(E)=30`
`n(A)=50,n(L)=70`
and `n[(EnnAnnL')uu(EnnA'nnL)uu(E'nnAnnL)=44`
or `n(EnnAnnL')+n(EnnA'nnL)+n(E'nnAnnL)=44` [`because` each case is mutually exclusive]
or `n(EnnA)-n(EnnAnnL)+n(EnnL)-n(EnnAnnL)+n(EnnL)-n(EnnAnnL)=44`

`impliesn(EnnA)+n(EnnL)+n(AnnL)-3n(EnnAnnL)=44" ... (i)"`
`because n(EuuAuuL)=100`
`therefore n(E)+n(A)+n(L)-n(EnnA)-n(AnnL)-n(EnnL)+n(EnnAnnL)=100`
`implies30+50+70-{44+3n(EnnAnnL)}+n(EnnAnnL)=100` [from Eq. (i)]
`implies 6-2n(EnnAnnL)=0`
`therefore n(EnnL nnA)=3`
Hence, there are three claims involved in loss or injury to all the three.