Of the 200 graduate students who were interviewed for a par time position at a call center, 110 had a bicycle, 25 had a master card and 130 had a mobile phone. 50 of them had both, a bicycle and a master card, 30 had both, a master card and a mobile phone and 60 had both, a bicycle and mobile phone and 10 had all three. How many candidates had none of the three?

65 candidates.

Step-by-step explanation:

From the above question, we have the following information

Total number of graduate students = n(B ∪ MP ∪ MC)

200 students

Number of students with bicycle = n(B) = 110

Number of students with MasterCard = n(MC) = 25

Number of students with Mobile phone = n(MP) = 130

Number of students with both bicycle and Master card = n ( B ∩ MC) = 50

Number of students with both Mastercard and Mobile phone = n ( MC ∩ MP) = 30

Number of students with both Bicycle and Mobile phone = n ( B ∩ MP) = 60

Number of students that had all three =

n ( B ∩ MP ∩ MC) = 10

Number of candidates that had none of the three = ( None of B ∩ MP ∩ MC)

Since this is a three element set, the formula is given as:

n(B ∪ MP ∪ MC) = n(B) + n ( MP) + n (MC) - n ( B ∩ MP) - n ( B ∩ MC) - n ( MC ∩ MP) + n (B ∩ MP ∩ MC) + ( None of B ∩ MP ∩ MC)

n(B ∪ MP ∪ MC) - ( None of B ∩ MP ∩ MC) = n(B) + n ( MP) + n (MC) - n ( B ∩ MP) - n ( B ∩ MC) - n ( MC ∩ MP) + n (B ∩ MP ∩ MC)

200 - ( None of B ∩ MP ∩ MC) = 110 + 25 + 130 - 60 - 50 -30 + 10

200 - ( None of B ∩ MP ∩ MC) = 110 + 25 + 130 + 10 - 60 - 50 - 30

= 200 - ( None of B ∩ MP ∩ MC) = 275 - 140

= 200 - ( None of B ∩ MP ∩ MC) = 135

( None of B ∩ MP ∩ MC) = 200 - 135

( None of B ∩ MP ∩ MC) = 65

Therefore, 65 candidates had none of the three