**MATH 425-4: EXTRA CREDIT PROBLEMS II**

Math 425-4, Fall 1997

Prof. Hochster

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** 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]

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