A standard Alberta license plate has three letters followed by three or four digits.
(a)
How many different vehicles can the province license with this scheme?
(b)
Do you think the province was right to expand license plates by adding another digit, or do you think it should have added another letter instead? (Or, as a third possibility, is it irrelevant in practical terms?)
You roll a six-sided die ten times. How many different sequences of rolls are possible?
(b)
Describe how Task a relates to the problem of determining \(\card{\words{\Sigma}_{10}}\) for a suitable alphabet \(\Sigma\text{.}\)
Activity20.3.
Let \(\Sigma = \EngAlphabet\text{.}\) How many words in \(\words{\Sigma}_5\) end in the letter \(z\text{?}\) How many do not?
Activity20.4.
You and your five housemates pick names out of a hat each week to determine who is going to clean the toilet. Over a three-week period, how many different sequences of toilet bowl cleaners could be determined in this fashion
if names are placed back in the hat after each draw?
if names are not placed back in the hat after each draw?
Activity20.5.
How many natural numbers between \(1\) and \(1,000,000\) (inclusive) contain the digit \(5\text{?}\)
You have a list of the names of twenty students. Ten of the students are domestic students and the other ten are out-of-province students. How many students must you select from the list to be certain to form a group that contains at least one domestic student and at least one out-of-province student?
Activity20.9.
Let \(n\) be a fixed natural number. Determine the smallest number \(M\) for which the following statement is true: every subset of
You’re cleaning up your little nephew’s toy room. There are \(T\) toys on the floor and \(n\) empty toy storage boxes. You randomly throw toys into boxes, and when you’re done the box with the most toys contains \(N\) toys.
(a)
What is the smallest that \(N\) could be when \(T = 2 n + 1\text{?}\)
(b)
What is the smallest that \(N\) could be when \(T = k n + 1\text{?}\)
(c)
Now suppose that the number of toys \(T\) satisfies
\begin{equation*}
T \lt \frac{n (n - 1)}{2} \text{.}
\end{equation*}
Prove that when you are done cleaning up, there will be (at least) one pair of boxes that contain the same number of toys.