White or Wong?

A man is mailing out five passports to five different people. There are five envelopes on the table, each bearing the name of the owner of a passport. "Imagine I were just to stick the passports into envelopes at random," the man thinks. "I wonder how many ways I could commit the perfect blunder and put every passport in the wrong envelope?" How many ways can this be done?




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SOLUTION:


Imagine we put one of the passports in the correct envelope, and then put all the other passports in every possible combination of envelopes. There would be 5*4!=120 ways of doing this. Now let's imagine we put two in the correct envelope and run through every combination for the other three. There are C⁵₂*3!=60 ways of doing this. Now imagine we put three in the correct envelope and ran through the possible combinations for the remaining pair. There are C⁵₃*2!=20 ways of doing this. Now, imagine we put four passports in the correct envelope, and allow the fifth envelope to go in any remaining envelope. There are C⁵₄*1!=5 ways this can be achieved. Placing five passports in the correct envelope obviously only be done in 1 way.

A little thought reveals that the total number of ways the passports could all be placed in the wrong envelope is given by

N=120-120+60-20+5-1=44.

So the man can place all the passports in the wrong envelopes in 44 different ways.