The formula tells us that the entries on row n of the Pascal Triangle read the same left to right as they do right to left. The combinatorial significance of this formula is that the number of ways of choosing k elements from a set of n elements is equal to the number of ways of omitting n - k of the elements.

A problem analogous to that of finding the coefficients of a binomial expansion is that of finding the coefficents in
 

These coefficients, called the trinomial coefficients, are naturally more complicated but, fortunately, can be expressed in terms of the binomial coefficients in the following way. Let us look for the coefficient of x⁶y³z¹ in (x + y + z)¹⁰. From this product of ten factors we must choose six x's, three y's, and one z. We can choose six x's from a set of ten in ways and three y's from the remaining four factors in ways, and the z from the remaining factor inway. Therefore the trinomial coefficient of x⁶y³z¹ in (x + y + z)¹⁰ can be written asor, since as  We can obtain an alternate representation of this number, however, by choosing the z first in ways, then the six x's from the remaining nine, and finally the three y's from the remaining three. Thus the coefficient would appear as or  Hence  By choosing the y's first, one can see that this coefficient could also be expressed as The reader may find other forms of the coefficient.
 
 

Tuesday, May 12, 1998