Suppose we define a new “product” a*b of two numbers a and b as follows :  

                                  a*b = ab2.

 Notice that this product is not commutative:

                         a*b g b*a, in general, since ab2 g ba2, in general;

 and also it is not associative : 

That is, a*b*c has two meanings according to where we insert brackets:

 (a*b)*c = (ab2)*c = ab2c2, whereas a*(b*c) = a*(bc2) = a(bc2)2 = ab2c4.

 The Problem :

 How many possible meanings does     a  *b  *c * d * e “            have?               

That is, in how many ways can we insert brackets ?

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Last modified: June 18, 2007