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Listing Possibilities

Sometimes the only real way to solve an arrangements problem is to list all the possible arrangements. The problem may have some limited symmetry to reduce the effort involved, but the symmetry may be broken in a way that would make it unsafe to rely purely on calculations.

A good example is the following.

Four men and three women stand in a line. One of the men is married to one of the women and they insist on standing together. How many arrangements are possible?

Call the four men A, B, C and M, and call the three women W, Y and Z.

M is married to W.

The men can be arranged ABCM and the women can be arranged WYZ.

M must stand next to W so we could have

ABCMWYZ

We can treat MW as a single unit so copuld also have

MWABCYZ, AMWBCYZ, ABMWCYZ, ABCYMWZ and ABCYZMW

In addition M and W could swap places which would double the number of arrangements.

So far there are 6*2=12 arrangements.

The other men and women can all stand anywhere. No man need stand next to or avoid any particular man or woman and no woman need stand next to or avoid any particuloar woman, so the remaining three men and two women can be thought of as a single group of five objects, with 5!=120 possible arragements.

Then there are 12*120=1440 possible arrangements of the group with the man standing next to his wife.

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