Twenty four people are part of three committees which are to look at research, teaching, and administration respectively. No two committees have any member in common. No two committees are of the same size. Each committee has three types of people: bureaucrats, educationalists, and politicians, with at least one from each of the three types in each committee. The following facts are also known about the committees:

- The numbers of bureaucrats in the research and teaching committees are equal, while the number of bureaucrats in the research committee is 75% of the number of bureaucrats in the administration committee.
- The number of educationalists in the teaching committee is less than the number of educationalists in the research committee. The number of educationalists in the research committee is the average of the numbers of educationalists in the other two committees.
- 60% of the politicians are in the administration committee, and 20% are in the teaching committee.

a) The size of the research committee is less than the size of the teaching committee

b) In the teaching committee the number of educationalists is equal to the number of politicians

c) The size of the research committee is less than the size of the administration committee

d) In the administration committee the number of bureaucrats is equal to the number of educationalists

a) The size of the teaching committee

b) The size of the research committee

c) The total number of educationalists in the three committees

d) The total number of bureaucrats in the three committees

Let the number of bureaucrats in the research committee = x and total number of politicians = y

From point 1) Then number of bureaucrats in the teaching and administration committees will be x and 4x/3 respectively.

From point 3) number of the politicians in the administration committee teaching committee will be 0.6y and 0.2y respectively.

Now let Number of educationalist in research , teaching and administration department are a, b and z such that a > b and a = (b+z)/2 thus z > a > b and a+b+z = 3a

From the given information following table can be made :

Research | Teaching | Administration | Total | |

Bureaucrats | x | x | 4x/3 | 10x/3 |

Politicians | 0.2y | 0.2y | 0.6y | y |

Educationalist | a > b | b | z | 3a |

Now as all the values in the table must be integers so minimum value of x = 3 and y = 5 also

10x/3 + y + 3a = 24

So the only possible value for a, x and y are 3, 3 and 5 . Thus the table with required number of people in each committees will be

Research | Teaching | Administration | Total | |

Bureaucrats | 3 | 3 | 4 | 10 |

Politicians | 1 | 1 | 3 | 5 |

Educationalist | 3 | 2 | 4 | 9 |

Total | 7 | 6 | 11 |

Or

Research | Teaching | Administration | Total | |

Bureaucrats | 3 | 3 | 4 | 10 |

Politicians | 1 | 1 | 3 | 5 |

Educationalist | 3 | 1 | 5 | 9 |

Total | 7 | 5 | 12 | 24 |

Now all the question can be answered :

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