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By Gerald Berman and K. D. Fryer (Auth.)

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3. 3 to obtain /(5),/(6),/(7), and/(8) for the polynomial f(x) (b) (c)c 4. = x 4 - 4JC3 + 6x2 - 3x. Set up similar tables for the polynomials in (ii) and (iii) of 1(a) and use these tables to evaluate/(5),/(6),/(7), and/(8) for these poly­ nomials. 3 and evaluate f(x) for x = 5, 6 , . . , 100 for the three polynomials of Problem 1. A monic polynomial of degree n is one in which the coefficient of xn is + 1 . (a) (b) (c) Show that A2f(x) = 2 iff(x) is any monic quadratic polynomial in x. Show that A3f(x) = 6 if f(x) is any monic cubic polynomial in x.

The first line contains the origin; the second, the two lattice points (0, 1) and (1,0); the third, the three lattice points (0, 2), (1, 1), (2, 0); etc. We may list the number of paths to the lattice points in each member of this family of parallel lines in the following pattern : 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 and so on. Each entry is the sum of the two above it to the right and to the left. This follows because of the above result that the number of paths to the point (#, b) is the sum of the number of paths to the points {a — 1, b) and (a, b - 1).

In how many ways can he make his selection? 5. If the student in Problem 4 is to choose four questions from the first seven and six from the last eight, in how many ways can he make his selection? 44 2 PERMUTATIONS AND COMBINATIONS 6. Twenty-four points, no three collinear, lie in a plane. How many line segments can be formed having these points as terminal points ? How many triangles can be formed having these points as vertices? 7. In how many ways can a committee of four be selected from six men and eight women if the committee must contain at least two women and if Mrs.

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