21.5: Activities
If you know what the choose function is, for this activity set pretend that you don't.
Write down all permutations of the set \(A = \{c,a,t\}\text{.}\) Express your permutations as functions from \(A\) to itself.
Write down some example permutations of size \(3\) from the set \(A = \{t,r,u,c,k\}\text{.}\)
Verify the equality
\begin{equation*} \dfrac{(n+1)!}{(k+1)!(n-k)!} = \dfrac{n!}{k!(n-k)!} + \dfrac{n!}{(k+1)!(n-k-1)!} \text{.} \end{equation*}
A child has the following set of refrigerator magnets: \(\{A,B,C,D,E,F,G,H,I,J\}\text{.}\)
- How many four-letter words can the child form? (Nonsense words allowed.)
- How many five-letter words can the child form if the middle letter must always be a vowel?
- If the child were able to form one word per second, and never stopped to eat or sleep, how many days would it take to form every possible word that uses all of the magnets?
- How many ways could student groups have been formed today if both group membership and group station location matter? (But assume that each station always has the number of students it has now.)
- How many ways could student groups have been formed today if only group membership matters? (Again assume that each group station always has the number of students it has now.)
- How many binary words of length \(10\) contain at least two zeros?
- How many binary words of length \(10\) contain at least at least three ones?
Consider the letters in the word \(PEANUT\text{.}\)
- How many six-letter words can be formed using these letters? (Each letter can only appear once.)
- How about if the vowels must be at the beginning?
- How about if no consonant may be isolated between two vowels?
You're cleaning up your shop and it's time to hang all your screwdrivers in a row on your pegboard. You have two slot-head screwdrivers, three Phillips-head screwdrivers, and four Robertson-head screwdrivers. (Assume that screwdrivers of the same type are of different sizes.)
- In how many different orders can you arrange your screwdrivers?
- How about if all the slot-heads are arranged on the left, all the Phillips-heads in the middle, and all the Robertson-heads are arranged on the right?
- How about if all screwdrivers of a particular type are arranged together, but the types are arranged in no particular order?
You're cleaning up your shop and it's time to hang all your screwdrivers in a row on your pegboard. You have five screwdrivers of each type: slot-head, Phillips-head, and Robertson-head. (Assume that screwdrivers of the same type are all of different sizes.)
- In how many different orders can you arrange your screwdrivers if the types must alternate: first slot-head, then Phillips-head, then Robertson-head, then slot-head, then Phillips-head, then ….
- How about if the types must alternate, but with no restriction on the order of the types?
- How many ways are there to arrange six people in a circle?
- How about if there are two people who cannot sit beside each other?
- How about if there is one person who cannot sit directly to the right of some other person?
- How many ways are there to arrange three professors and three students in a circle so that professors and students alternate?
- Answer the same question for \(n\) professors and \(n\) students.
How many ways could you choose numbers \(a,b,c\) from the set \(\mathbb{N}_{<11}\text{,}\) allowing repetition, so that the sum \(a+b+c\) is at least 5?