A large logistic manufacturing company has 100 employees specializing in different machines. From them, 20 are proficient in machine P , 45 in machine Q , 30 in machine R , 1 in machine P and Q , 6 in machine Q and R , 5 in machine R and P , and just 1 specialist in all three machines. i) Determine how many of them only specialize in machine Q. ii) And how many of them do not specialize in any of the machines.

we first define the following

N=100, N(P) =20, N(Q) =45, N(R) =30 , N(P and Q)=1, N(Q and R)=6, N(R and P)=5, N(P and Q and R) = 1

Hence we solve part i) and ii) as below

i) The number of those specializing in machine Q only will be

N(Q) - N(P and Q) + N(P and Q and R) - N(Q and R)

= (45 - 1 +1 - 6) = 39 employees

ii)The number of those who do not specialize in any of the machines equals

N - [ N(P) +N(Q) + N(R) - N(P and Q) - N(P and R) - N(Q and R) + N(P and Q and R) ]

= 100 - [20 +45 +30 -1 -5 - 6 +1]

= 100 - 84

= 16 employees