Roorkee University has to send 10 professors to 5 centers for its entrance examination, 2 to each center. Two of the enters are in Roorkee and the others are outside. Two of the professors prefer to work in Roorkee while three prefer to work outside. In how many ways can this be made if the preferences are to be satisfied?

Answer

5400

Explanation

We have to select 4 professors for Roorkee and 6 professors for outside. Again, 2 professors prefer Roorkee and 3 outside, so we are left with 5 professors.
The number of ways in which two more professors for Roorkee can be selected is ` .^(5)C_(2)=10`.
And remaining 3 professors are left for outside. Now, 6 professors outside Roorkee can be allotted to 3 centres in `6!//(2!2!2!3!)xx3!` ways.
Now, 4 professors for 2 centers in Roorkee can be allotted in `4!//(2!3!3!)xx2!` ways. hence, the total number of ways is
`10xx(6!)/(2!2!2!3!)xx3!xx(4!)/(2!2!2!)xx2!=5400`