A survey was taken among 100 people to find their preference of watching T. V. programs. There are 3 channels. Given the no of people who watch

at least channel 1
at least channel 2
at least channel 3
no channels at all
at least channels 1and 3
at least channels 1 and 2
at least channels 2 and 3
Find the no of people who watched all three.

• Consider the standard Venn diagram consisting of three circles mutually intersecting, each circle representing the set of people watching each channel. Let these be A (channel 1), B (channel 2) and C (channel 3).
  
  Name the portions in the figure as follows:
  The portion in A alone: a
  The portion in B alone: b
  The portion in C alone: c
  The portion common to B and C only: d
  The portion common to C and A only: e
  The portion common to A and B only: f
  The portion common to A, B and C: g
  
  Now, these circles are all enclosed in a rectangle representing the entire population. The portion inside the rectangle, but outside all the circles be called r.
  
  Given:
  1. at least channel 1 : a+f+g+e
  2. at least channel 2 : b+f+g+d
  3. at least channel 3 : c+e+g+d
  4. no channels at all : r
  5. at least channels 1 and 3 : e+g
  6. at least channels 1 and 2 : f+g
  7. at least channels 2 and 3 : d+g
  
  To find: no of people who watched all three: g
  
  Given, a+b+c+d+e+f+g+r = 100
  So, a+b+c+d+e+f+g = 100-r
  This can be calculated, since r is given.
  
  Substituting (5.) in (1.), (6.) in (2.) and (7.) in (3.), we can calculate a+f, b+d and c+e.
  Adding all these, we get a+b+c+d+e+f
  
  Now, g = 100 - r - (a+b+c+d+e+f)
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