By Abraham P Hillman
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Extra resources for Algebra through problem solving
That is, we can associate the term 1 with the empty subset of S; the terms x1, x2, and x3 with the singleton subsets of S; the terms x1x2, x1x3, and x2x3 with the doubleton subsets; and x1x2x3 with S itself. ) Next we replace each of x1, x2, and x3 by x in our two expressions for M. This results in (1 + x)3 = 1 + 3x + 3x2 + x3. Thus we see that 3 k for k = 0, 1, 2, 3 is the number of ways of choosing a subset of k elements from a set S of 3 elements. Similarly, one can see that the number of ways of choosing k elements from a set of n elements is n .
We have seen that binomial coefficients, Fibonacci and Lucas numbers, and factorials may be defined inductively, that is, by giving their initial values and describing how to get new values from previous values. Similarly, one may define an arithmetic progression a1, a2, ... , t - 1. Then the values of a1 and d would determine the values of all the terms. A geometric progression b1, ... , bt is one for which there is a fixed number r such that bn+1 = bnr for n = 1, 2, ... , t - 1; its terms are determined by b1 and r.
What is the sum of all the trinomial coefficients in (x + y + z)100? 18. What is the sum of the coefficients in each of the following: (a) (x + y - z)100? (b) (x - y + z - w)100? 19. List the even permutations of 1, 2, 3, 4. 20. List the odd permutations of 1, 2, 3, 4. R 21. Let P be a permutation i, j, h, ... k of 1, 2, 3, ... , n. (a) Show that if i and j are interchanged, P changes from odd to even or from even to odd. (b) Show that if any two adjacent terms in P are interchanged, P changes from odd to even or from even to odd.