Mini-Quiz June 7

--- tags: 225-spr22, mth225, lt-quiz --- # Mini-Quiz June 7 :::info This is a **mini-quiz**, designed to give you an attempt at **a small number of Learning Targets** of your choice from among the more recent ones we've covered, especially CORE targets, done **quickly** and **in class**. This mini-quiz contains problems for **Learning Targets 7, 13, and 14**. You may do any combination of these you want. Or if you prefer to practice some more and try again this week, do nothing and just take the main Quiz on Thursday. Work the problems you select out on paper or in a notes app, export the work to PDF (one PDF per Learning Target) and upload it to the appropriate assignment folder on Blackboard just as in the case of a full Quiz. **Your work is due by 2:30pm.** No submissions will be accepted after that time. ::: ## Learning Target 13 :::warning (**CORE**) I can apply the Additive and Multiplicative Principles and the Principle of Inclusion/Exclusion to formulate and solve basic combinatorics problems. ::: *Work out the answer to each of the counting questions below. State the answer clearly, and show all work and explain your reasoning in words on each. **Note:** Answers with insufficient explanations or work are not accepted.* In the country of Boratistan, telephone numbers consist of five letters (selected from the English alphabet, A through Z) and five digits (selected from 0 through 9). How many telephone numbers are possible... 1. In all, if repetitions of letters or digits are allowed? 2. In all, if repetitions of letters or digits are *not* allowed? 3. That either have "ZZZZZ" as their five letters or have 0 as the final digit? (Repetitions of letters and digits are allowed.) **Success criteria:** All three parts have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least two of the three answers must be correct and sufficiently justified; the third may be incorrect as long as it results from a "simple" error. ## Learning Target 14 :::warning (**CORE**) I can compute a binomial coefficient and apply the binomial coefficient to solve basic combinatorics problems. ::: 1. Compute the exact numerical value of the following. Either show work or explain in one sentence how you got your answer. Leave no fractions in your answers. (a) $\binom{12}{5}$ (b) $\binom{12}{12}$ (c) $\binom{12}{1}$ 2. Solve the following counting problems. State your answer clearly and justify each answer with math work or a brief verbal explanation. The justification for each answer must involve the binomial coefficient. (a) The English alphabet has 26 letters (`A` through `Z`). How many 10-letter subsets of the alphabet are there? (b) How many 10-bit strings are there that have exactly 7 "1" bits? **Success criteria:** All three of the answers on the first item are correct and have sufficient justifications. Both parts of the second item have good-faith efforts at a correct answer and contain clear, substantive explanations that justify the approach and answer. At least one of the two answers must be correct and sufficiently justified; the other may be incorrect as long as it results from a "simple" error. ## :new: Learning Target 17 :::warning (**CORE**) I can generate several values of a sequence given either a closed-form or recursive definition. ::: List the first six (6) terms of each of the following sequences. You do not need to show your work, but your answers must be correct. 1. $a_n = 2(3^n) + 3, \ \text{where} \ n = 1, 2, 3, \dots$ 1. $b_n = 3 - 2n \ \text{where} \ n = 0, 1, 2, \dots$ 1. $c_0 = 2, \ \text{and} \ c_n = 2c_{n-1} + 3n \ \text{if} \ n > 0$ 1. $d_0 = 1, d_1 = 2 \ \text{and} \ d_n = 7d_{n-1} - 10d_{n-2} \ \text{if} \ n > 1$ **Success criteria:** At least three of the four sequences have all six terms correctly listed.