Math 425-4, Fall 1997
Prof. Hochster
EC#5. After an in vitro fertilization procedure (IVF) four fertilized eggs are placed in the mother's uterus. Assume that each egg has one chance in ten of implanting successfully, and that this is independent of whether any of the others implants. What is the probability that none of the eggs will implant? That a single egg will implant? That there will be twins? That there will be triplets? That there will be quadruplets? [9/17]
EC#6. Give a formula for the number of ordered triples (A, B, C) of subsets of a set with n elements such that A and B are disjoint and B and C are disjoint as well. Explain why your formula is correct. [9/19]
EC#7. Let x, y and n be nonnegative integers. Suppose that you are interested in the number of strings of length n constructed from an ``alphabet" with x+y characters, where x of the characters are letters and y are numerals. On the one hand, the number of such strings should be x+y raised to the n th power. (Explain why.) A different way of counting is the following. For every integer r from 0 to n, count the number of strings in which r of the characters are letters and n-r are numerals, and then add all these terms up. Carry through this idea, and then explain why your results give another proof of the binomial theorem, at least when x and y are nonnegative integers [9/22]
Remark. The idea of problem #7. actually proves the binomial theorem in general, because if two polynomials in one or several variables are equal for all values of the variables that are nonnnegative integers then they are equal. (This is true even if the coefficients are complex numbers.) This can be proved by induction on the number of variables. In fact, for polynomials in one variable x, if two polynomials f(x) and g(x) are equal for d+1 values of x, and both polynomials have degree at most d, then they are equal. (Otherwise their difference would be a nonzero polynomial of degree at most d with more than d roots.)
EC#8. Show that if two polynomials in two variables are equal for all nonnegative integer values of the variables, then they are equal. (You may assume the case of one variable, by the discussion in the remark.) [9/26]
EC#9. Generalize the idea of problem #7. to prove the multinomial theorem for nonnegative integer values of the variables. [9/29]