The crew of a rowing team of 8 members is to be chosen from 12 men (m1, m2, …., m12) and 8 women (w1, w2,…., w8), such that there are two rows, each row occupying one the two sides of the boat and that each side must have 4 members including at least one women. further it is also known w1 and m7 must be selected for one of its sides while m2, m3 and m10 must be selected for other side. what is the number of ways in which rowing team can be arranged.
Answers
Given 12 men and 8 women and the places to be filled are 8 , four on each side
with atleast one woman on each side
the side in with M2, M3 and M10 are seated requires 1 woman since atleast one woman has to be present
So 1 woman can be selected in 7 ways( since W1 is already allotted to other side)
The four places can be arranged by this side in 4! ways
So together the combination for this side is 7×4! ways
The remaining members are 20- (4+2) = 14 members
4 members are already allotted to the second side and w1 and M7 are allotted
The two places can be allotted in ₁₄C₂ ways
The four places can be combined in 4! ways
So the total number of combination is ₁₄C₂× 4!
The total number of combinations are 7x4!x₁₄C₂×4!
the members of both the sides can be arranged in a total of 2×7x4!x₁₄C₂×4! ways