Problem 5. How many ways are there to choose a dozen donuts from 20 varieties 5(a) if there are no two donuts of the same variety? 5(b) if all donuts are of the same variety? 5(c) if there are no restrictions?

Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) For the rest of the problems in this section, decide whether to use permutations or combinations. 4.) A bag of marbles contains 3 yellow, 4 white, and 9 blue marbles. (a) How many samples of 2 marbles can be drawn? (b) How many samples of 4 marbles can be drawn? (c) How many samples of 4 marbles can be drawn in which exactly 2 are blue?

6 people consisting of 3 couples are seated in a row of 6 chairs to take group pictures.(a) In how many ways could they be seated?(b)How many ways could they be seated if each couple should sit next to each other?Justify your answer.(c) One of the couples broke up and do not want to sit next to each other. Whereas the other 4 want to sit next to their partners. In how many ways could they be seated?

2. A student is to answer 10 out of 13 questions. Find the number of choices where the student must answer: (a) the first three questions. (b) first or second question but not both. (c) exactly 2 out of first 5 questions. (d) at least 4 out of the first 6 questions.

Note: In this problem set, the notation C(n, k) is used for the combination "n choose k." Alternate notation: ( ) 16.) A group of 2 professors and 5 students (3 grad students and 2 undergrads) go to a movie together. The professors want to sit together, and the grad students want to sit together. How many ways are there to seat this group in a row of 7 seats?

A grocery store is placing an order for dairy products. There are 8 varieties of dairy products sold in cases, where each case contains all the same variety of dairy products. The store will order 50 cases in total. (a) How many ways are there to place the order? (b) One of the varieties of dairy products is Milk. How many ways are there to place the order if they order at most 20 cases of Milk?

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?

(a) A company has  31 salespeople. A board member at the company asks for a list of the top 4  salespeople, ranked in order of effectiveness. How many such rankings are possible?   (b) From the  13 albums released by a musician, the recording company wishes to release  10 in a boxed set. How many different boxed sets are possible

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.

In how many ways can 10 engineers and 4 doctors be seated at a round table if all the 4 doctors do not sit together? 14! 10! x 4! 13! - (10! x 4!) 13! x 4!