Can you help me in answering these questions on the slides as well as analyzing them. Thanks so much. :)

\fSOME WARM -UP QUESTIONS Discuss the following counting questions about the four friends Alicia, Benoit, Clara, and Deepak. 1. In how many ways can a Snickers, an Almond Joy, and a Kit Kat be distributed among the friends? 2. In how many ways can the friends be arranged in a lunch line? 3. Having got their lunches, in how many different ways can the four friends sit at a circular lunch table for four? 4. After lunch the four friends need to study. In how many ways can they pair up into two study groups of 2?THE PRODUCT RULE (DON'T WORRY, NOT THE ONE FROM CALCULUS!) To count the number of sequences of length 1, where the first term is chosen from set A, the second term is chosen from set A,, and so on, we simply multiply together the sizes of these sets: # sequences = [Al| X |A2| x .. * |All Example 1. This is the candy bar situation: there are 4 friends to whom the Snickers con be given; the some 4 to whom the Almond Joy can be given; and the some 4 to whom the Kit Kat con be given: # ways = 4 X 4 X 4 = 64 Example 2. The product rule also counts the 2-dice sample space: there are 6 possible outcomes for the blue die and 6 for the red die. Altogether there are 6 x 6 = 36 combined outcomes.PERMUTATIONS: WHEN ORDER MATTERS Example. Candy bars again. What if no friend can receive more than one candy bar? How does that change the count? There are still 4 friends to whom the Snickers can be given; but now only 3 to whom the Almond Joy can be given; and then 2 to whom the Kit Kat can be given. Altogether, # ways = 4 X 3 X 2 = 24 We call each choice a 3-permutation of the four friends.Mil [IAMPLE A student society with 15 members wishes to choose officers: a president, a secretary, and a treasurer. In teams {3 minutes]: see if you can figure out the number of different woys this can be done. Exercise: ten you write down a general expression (formula) for the number of rpermutations of a set of n elements? Let's denote this number PEn,r]. WHEN DOES ORDER NOT MATTER? Example. The same 15-member student society again. What if they simply wanted to select a 3-person committee with no specific roles? In how many different ways can this be done? Is that the same number as before? If so, why? If not, why not? In teams (2 minutes): discuss!POKER HANDS (THE PROBLEMS FROM LAST TIME) The game of poker throws up a lot of nice counting problems. You don't need to have played poker to think about these problems. Here's what you need to know: There are 52 cards in a deck. A poker hand is made up of 5 cards. 4 suits: . (spades), @ (hearts), + (clubs), o (diamonds) 13 ranks: A(Ace) , 2, 3, 4, 5, 6, 7, 8, 9, 10, J(Jack), Q(Queen), K(King) Take 5 minutes in your teams to discuss: 1. How many different poker hands are there in total? 2. How many of those contain cards all of the same suit? 3. How many form a sequence, e. g. 03 +4 05 46 70 ? 4. How many have three cards of the same rank, e.g. 06 46 06 42 5. What other types of poker hand could we try to count?BINOMIAL COEFFICIENTS In the last couple of examples we have essentially been counting the number of r-element subsets of a set of n elements. This number comes up a lot in mathematics: C(n, r) = P(n, r) + r! = ( n! / (n-r)! ) + r! Exercise: Think about the expansion of (1 + x)ll Can you determine the coefficient of x4? Does your answer relate at all to the symbol C(m, r)