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 Math Central Quandaries & Queries
 Question from Cecilia, a student: If the letters G, H, S are used exactly 3 times how many 9 letter combinations can be generated?

Cecilia,

Suppose all 9 letters are distinguishable, perhaps by colour. Assume there is a red, blue and green copy of each letter. Since you now have 9 distinct letters there are 9! permutations of these letters. Here for example are three of them.

G S S G H G H H S
G S S G H G H H S
G S S G H G H H S

What happens if the Gs are no longer distinct? Suppose they are all black then the 3 permutations above become

G S S G H G H H S
G S S G H G H H S
G S S G H G H H S

The three permutations are now indistinguishable. In fact for every arrangement of the Hs and Ss, for example

S S    H    H H S

there are 3! ways to place the 3 Gs. Hence if the 3 Gs are indistinguishable then there are only 9!/3! permutations.

What happens now if the Hs are all black?

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.