SWIM 2007 Math 6105
Throughout we use both the notations ๎n
r๎and Cn
rfor the number n!
(nโr)!r!.
1. A falling number is an integer whose decimal representation has the property
that each digit except the units digit is larger than the one to its right. For
example 96521 is a falling number but 89642 is not. How many n-digit falling
numbers are there, for n= 1,2,3,4,5,6,7,8,and 9? What is the total number
of falling numbers of all sizes?
2. Cyprian writes down the middle number in each of the ๎9
5๎= 126 five-element
subsets of S={1,2,3,4,5,6,7,8,9}. Then he adds all these numbers together.
What sum does he get?
3. Counting sums of subset members.
(a) How many numbers can be expressed as a sum of two or more distinct
members of the set {1,2,3,4,5,6,7,8,9}?
(b) How many integers can be expressed as a sum of two or more different
members of the set {0,1,2,4,8,16,32}?
(c) How many numbers can be expressed as a sum of four distinct members
of the set {17,21,25,29,33,37,41}?
(d) How many numbers can be expressed as a sum of two or more distinct
members of the set {17,21,25,29,33,37,41}?
(e) How many integers can be expressed as a sum of two or more distinct
elements of the set {1,โ3,9,โ27,81,โ243}?
4. How many of the first 242 positive integers are expressible as a sum of three or
fewer members of the set {30,31,32,33,34}if we are allowed to use the same
power more than once. For example, 5 = 3 + 1 + 1 can be represented, but 8
cannot. Hint: think about the ternary representations.
5. John has 2 pennies, 3 nickels, 2 dimes, 3 quarters, and 8 dollars. For how
many different amounts can John make an exact purchase (with no change
required)?
6. How many positive integers less than 1000 have an odd number of positive
integer divisors?
7. An urn contains marbles of four colors: red, white, blue, and green. When
four marbles are drawn without replacement, the following events are equally
likely:
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