# Inclusion-Exclusion Principle - Problem

There was a question in my exam discrete maths that I just couldn't figure out. I know it's supposed to be solved using the inclusion-exclusion principle. anyone able to help me solve and understand this question.

For a survey, 200 people are asked about which forms of transport the had used in the last month it was found that

150 used trains,

80 had cycled and used trains, and

180 had used one or the other of these two forms of transport.

Question: How many people had cycled in the last month?

Question: How many had not used either form of transport.

• Welcome to MathSE. What have you tried? – jvdhooft Apr 10 '18 at 9:12
• See Inclusion–exclusion principle for the first Question. Use subtraction for the second one. – Mauro ALLEGRANZA Apr 10 '18 at 9:15
• Do you know Venn diagrams? – Arnaud Mortier Apr 10 '18 at 9:20
• Yes, I know Venn Diagrams that's how I got my answer I'm not sure if it's correct. – user9455658 Apr 10 '18 at 9:24

LET $c$ and $t$ denote cycle and train.

$$c \cup t =180$$ $$c \cap t +80$$

$$t=150$$ $$180= c +150 -80$$

$$c=110$$

Those who did not use any of the two transportations are

$$200-180 =20$$

• Wow, thank you for that I get it now. – user9455658 Apr 10 '18 at 9:30
• Thanks for your attention and understanding. – Mohammad Riazi-Kermani Apr 10 '18 at 9:50