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Magazine Problem 3
How many lists of length 3 can be made from the symbols \(A, B, C, D, E, F\) if... (a) \(\ldots\) repetition is allowed. (b) ... repetition is not allowed. (c) \(\ldots\) repetition is not allowed and the list must contain the letter \(A\). (d) ... repetition is allowed and the list must contain the letter A.
Problem 3
A set \(X\) has exactly 56 subsets with 3 elements. What is the cardinality of \(X ?\)
Problem 4
This problem involves lists made from the letters \(T, H, E, O, R, Y,\) with repetition allowed. (a) How many 4 -letter lists are there that don't begin with \(T\), or don't end in \(Y ?\) (b) How many 4 -letter lists are there in which the sequence of letters \(T, H, E\) appears consecutively (in that order)? (c) How many 6 -letter lists are there in which the sequence of letters \(T, H, E\) appears consecutively (in that order)?
Problem 4
Select any five points on a square whose side-length is one unit. Show that at least two of these points are within \(\frac{\sqrt{2}}{2}\) units of each other.
Problem 5
Prove that any set of seven distinct natural numbers contains a pair of numbers whose sum or difference is divisible by 10 .
Problem 5
How many integers between 1 and 9999 have no repeated digits? How many have at least one repeated digit?
Problem 5
How many 16 -digit binary strings contain exactly seven 1's? (Examples of such strings include 0111000011110000 and 0011001100110010 , etc. )
Problem 5
This problem involves 8-digit binary strings such as 10011011 or 00001010 (i.e., 8-digit numbers composed of 0 's and 1 's). (a) How many such strings are there? (b) How many such strings end in \(0 ?\) (c) How many such strings have 1's for their second and fourth digits? (d) How many such strings have 1's for their second or fourth digits?
Problem 5
A bag contains 20 identical red balls, 20 identical blue balls, 20 identical green balls, and one white ball. You reach in and grab 15 balls. How many different outcomes are possible?
Problem 6
There are two 0's at the end of \(10 !=3,628,800 .\) Using only pencil and paper, determine how many 0's are at the end of the number \(100 !\)
