Question

Following an experiment in lab, you would like determine the number of arrangements possible for proteins adsorbing onto a surface. Consider 6 proteins adsorbing onto a 6 x 6 grid. The protein we're dealing with has long rigid 'arms'. As a result, once a protein adsorbs onto a site, no other protein can subsequently lie in the same row or column (diagonals are ok!) a) What is the total number of ways the 6 proteins can all adsorb onto the 6 x 6 grid? b) Generally, the first protein to adsorb will require the least amount of energy. Each event. What is the average energy for all 6 proteins adsorbing onto the surface? subsequent adsorption event can be approximated by E more energy than the previous

Answer 1

(a) since nothing is mentioned in the question it can be assumed that the proteins are not identical so, these 6 protiens can be rearranged in between themselves in 6! Ways for each of these, they can be absorbed in the 6 times 6 in k many where k is the number such that once a cell is selected, no subsequent cell can be in the same row or column the first cell can be any of the 36 cells After the first cell is selected, the remaining 5 cells of the corresponding column are not available any more so, the second cell can be selected in (36 - 1 - 1 - 5) = 25 After the second cell is selected the choice of the third cell will be further reduced to (25 - 1 - 4 - 1) = 16, since the row and column corresponding to the first cell is also unavailable along with those corresponding to the cell