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Using the digits 2 through 8 , find the number of different 5 -digit numbers such that: (a) Digits can be used more than once. (b) Digits cannot be repeated, but can come in any order. (c) Digits cannot be repeated and must be written in increasing order. (d) Which of the above counting questions is a combination and which is a permutation? Explain why this makes sense.

After another gym class you are tasked with putting the 14 identical dodgeballs away into 5 bins. This time, no bin can hold more than 6 balls. How many ways can you clean up?

Let \(A=\\{1,2,3, \ldots, 9\\}\) (a) How many subsets of \(A\) are there? That is, find \(|\mathcal{P}(A)|\). Explain. (b) How many subsets of \(A\) contain exactly 5 elements? Explain. (c) How many subsets of \(A\) contain only even numbers? Explain. (d) How many subsets of \(A\) contain an even number of elements? Explain.

hexadecimal We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different "digits" \(\\{0,1, \ldots, 9\\}\). Sometimes though it is useful to write numbers hexadecimal or base 16\. Now there are 16 distinct digits that can be used to form numbers: \(\\{0,1, \ldots, 9, \mathrm{~A}, \mathrm{~B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}\\} .\) So for example, a 3 digit hexadecimal number might be \(2 \mathrm{~B} 8\). (a) How many 2 -digit hexadecimals are there in which the first digit is \(E\) or \(F\) ? Explain your answer in terms of the additive principle (using either events or sets). (b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer? (c) How many 3 -digit hexadecimals start with a letter (A-F) and end with a numeral \((0-9) ?\) Explain. (d) How many 3 -digit hexadecimals start with a letter (A-F) or end with a numeral \((0-9)\) (or both)? Explain.

In an attempt to clean up your room, you have purchased a new floating shelf to put some of your 17 books you have stacked in a corner. These books are all by different authors. The new book shelf is large enough to hold 10 of the books. (a) How many ways can you select and arrange 10 of the 17 books on the shelf? Notice that here we will allow the books to end up in any order. Explain. (b) How many ways can you arrange 10 of the 17 books on the shelf if you insist they must be arranged alphabetically by author? Explain.

After gym class you are tasked with putting the 14 identical dodgeballs away into 5 bins. (a) How many ways can you do this if there are no restrictions? (b) How many ways can you do this if each bin must contain at least one dodgeball?

How many 9-bit strings (that is, bit strings of length 9 ) are there which: (a) Start with the sub-string 101? Explain. (b) Have weight 5 (i.e., contain exactly five 1 's) and start with the sub- string 101? Explain. (c) Either start with 101 or end with 11 (or both)? Explain. (d) Have weight 5 and either start with 101 or end with 11 (or both)? Explain.

Consider five digit numbers \(\alpha=a_{1} a_{2} a_{3} a_{4} a_{5},\) with each digit from the set \\{1,2,3,4\\} (a) How many such numbers are there? (b) How many such numbers are there for which the sum of the digits is even? (c) How many such numbers contain more even digits than odd digits?

Suppose you have sets \(A\) and \(B\) with \(|A|=10\) and \(|B|=15\). (a) What is the largest possible value for \(|A \cap B|\) ? (b) What is the smallest possible value for \(|A \cap B|\) ? (c) What are the possible values for \(|A \cup B|\) ?

A woman is getting married. She has 15 best friends but can only select 6 of them to be her bridesmaids, one of which needs to be her maid of honor. How many ways can she do this? (a) What if she first selects the 6 bridesmaids, and then selects one of them to be the maid of honor? (b) What if she first selects her maid of honor, and then 5 other bridesmaids? (c) Explain why \(6\left(\begin{array}{c}15 \\\ 6\end{array}\right)=15\left(\begin{array}{c}14 \\ 5\end{array}\right)\).