Solved: How many automobile registrations may the police

List all pennutations of four digits I, 2, 3, 4, taken all at a time.

List (a) all permutations, (b) all combinations without repetitions, (c) all combinations with repetitions, of 5 letters G, e, i, 0, 1I taken 2 at a time.

In how many ways can we assign 8 workers to 8 jobs (one worker to each job and conversdy)?

How many samples of 4 objects can be drawn from a lot of 80 objects?

In how many different ways can we choose a committee of 3 from 20 persons? First guess.

In how many different wa) s can we select a committee consisting of 3 engineers. 2 biologists. and 2 chemists from 10 engineers, 5 biologist~, and 6 chernists? First guess.

Of a lot of 10 items, 2 are defective. (a) Find the number of different samples of 4. Find the number of samples of 4 containing (b) no defectives, (c) I defective, (d) 2 defectives.

If a cage contains 100 mice, two of which are male, what is the probability that the two male mice will be included if 12 mice are randomly selected?

An urn contains 2 blue, 3 green, and 4 red balls. We draw I ball at random and put it aside. Then we draw the next ball, and so on. Find the probability of drawing at first the 2 blue balls, then the 3 green ones, and finally the red ones.

By what factor is the probability in Prob. 9 decreased if the number of balls is doubled (4 blue, etc.)?

Detennine the number of different bridge hands. (A bridge hand consists of 13 cards selected from a full deck of 52 cards.)

In how many different ways can 5 people be seated at a round table?

If 3 suspects who committed a burglary and 6 innocent persons are lined up, what is the probability that a witne~s who is not sure and has to pick three persons will pick the three suspects by chance? That the witness picks 3 innocent persons by chance?

(Birthday problem) What is the probability that in a group of 20 people (that includes no twins) at least two have the same birthday. if we assume that the probability of having birthday on a given day is 11365 for every day. First guess.

How many different license plates showing 5 symbols, nanlely, 2 letters followed by 3 digits. could be made?

How many automobile registrations may the police have to check in a hit-and-run accident if a witness reports KDP5 and cannot remember the last two digits on the license plate but is certain that all three digits were different?

CAS PROJECT. Stirling formula. (a) Using (7), compute approximate values of n! for 11 = I, ... ,20. (b) Detennine the relative error in (a). Find an empirical formula for that relative error. (c) An upper bound for that relative error is \(e^{1 / 12n} - L\). Try to relate your empirical formula to this. (d) Search through the literature for further infonnation on Stirling's fonnula. Write a short report about your findings, arranged in logical order and illustrated with numeric examples. Text Transcription: e^1/12n - L

TEAM PROJECT. Permutations, Combinations. (a) Prove Theorem 2. (b) Prove the last statement of Theorem 3. (e) Derive (11) from (8). (d) By the binomial theorem, \((a+b)^{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) a^{k} b^{n-k} \), so that \(a^{k}b^{n - k}\) has the coefficient \((_{k}^{n})\). Can you conclude this from Theorem 3 or is this a mere coincidence? (e) Prove (14) by using the binomial theorem. (f) Collect further formulas for binomial coefficients from the literature and illustrate them numerically. (a + b)^n = sum ^n _ k = 0 (_k^n) a^k b^n-k