4. In how many ways can you form a "word" (any sequence of letters, where each of the given letters must appear exactly once) from the letters: (a) method (b) combinactorial 1
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- You have one pile of 11 building blocks, each one a different color, and you also have another pile of 7 blocks, again each one a different color. How many ways are there to build a tower 4 blocks high from the first pile and another tower 3 blocks high from the second pile? Assume that the colors at each level of a tower matter. (A) 318 (B) C(11,4) C(7, 3) (C) P(11,4) P(7,3) (D) P(18,7) (E) 311 (F) C(18,7) (G) P(11,4)+ P(7,3) (H) C(11,4)+ C(7,3) O A OB OC O E O F OG H.Suppose there are 5 men and 8 women qualified to be members of a new committee of 6 people, includinga president, a vice-president, and a secretary.(a) How many different ways do you have to pick a president, vice-president and secretary if you need atleast 1 men and 1 women? Does the order matter? Why?(b) How many different committees can you form with at least 3 women? Does the order matter? Why?Your professor has 10 advanced statistics texts and 8 advanced calculus texts. (a) How many ways can the 10 statistics texts be arranged on the top shelf?(b) How many ways can the 8 calculus texts be arranged on the bottom shelf?(c) How many ways can the 18 texts be arranged together on a single shelf, with the statistics texts grouped together and the calculus texts grouped together?
- An instructor gives an exam with twenty questions. Students are allowed to choose any fifteen questions to answer. (a) How many different choices of fifteen questions are there? (b) Suppose nine questions require coding and eleven do not. (i) How many groups of fifteen questions contain seven that require coding and eight that do not? (ii) How many groups of fifteen questions contain at most eight that require coding? (c) Suppose the exam instruction specifies that at most one of questions 1 and 2 MAY BE included among the fifteen. How many different choices of fifteen questions are there? (d) Suppose the exam instruction specifies that either both questions 1 and 2 are to be included among the ten or neither is to be included. How many different choices of ten questions are there?How many ways can one select a subset of 10 elements from {1, 2, 3, . . . , 1000} so that the biggest and smallest elements selected differ by at most 100? Justify your answer.Suppose we want to choose 7 objects, without replacement, from 11 distinct objects.(If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices is taken into consideration? (b) How many ways can this be done, if the order of the choices is not taken into consideration?
- d. [Permutation] Solve the following problems - Find the number of words that can be formed by using the letters of the word "DEMOCRACY" where first letter will be always C and last letter will be R. e. [Combination] A group of 15 students is to be formed from 44 students of MATI00 Course. Among the students of the course there is 2 renowned batsmen, 3 spinners, 2 fast blowlers and 2 wicketkeepers. How many different ways this can be done so as always to (i) include 1 renowned batsman, 1 fast bowler, 1 spinner and 1 wicketkeeper; and (ii) exclude spinners? 2.Suppose we want to choose 6 objects, without replacement, from 10 distinct objects. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order of the choices does not matter? ☐ (b) How many ways can this be done, if the order of the choices matters?Fill in each blank and underline what to put in each box. Do not skip any blanks. if you do i will downvote.
- A classroom has 20 seats and 18 students. (a) How many different ways are there to select any 2seats from the 20 available? (b) How many ways are there for the 18 students to occupy the 18seats that remain after 2 are selected and set aside? (c) How many total ways are there for the 20students to occupy the 18 seats?2. I found some bingo cards in our classroom. A bingo card seems to have 5 rows and 5 columns. The center cell is blank, but the remaining cells each contain a number. No two cells contain the same number. In the first column, integers from 1 to 15 are used. In the 2nd column, integers from 16 to 30 are used, and so on. Thus, in the 5th column, integers from 61 to 75 are used. How many different bingo cards are there? Explain.