1. The following three prompta (regarding permutations, combinations, and the Binomial Theorem) are simple fill-in-the-blank prompts, where you're pulling words from the following word bank. You may use some words more than once, and you should NOT use all of the words in the bank. Word Bank: Addition Principle Multiplication Principle two r! n! ordered unordered algebraically equivalent binomial normal The fur n! (a) Explain why , P, = (n - p)! : by filling in the blanks: n Pr is the number of arrangements of objects taken from a act of size without replacement. By the . there are in ways to pick the first object, and then n - 1 ways to pick the second object, and so on until you've arranged a total of " objects, corresponding to " factors. Thus, . P+ = n . (n - 1) . (n -2) - -. (n - r + 1). This "falling factorial" is simply n! truncated after r factors, which is the expression (n - T (b) Explain why , C, = (n - rity! : by filling in the blanks: " , is the number of arrangements of objecta taken from a act of size without replacement. This is the number of r-permutations on a objects, compensating for the over-counting of equivalent permutations (which count as only one combination). Thus, ..C, = 2fr T ! because there are ways to arrange (permute) any given collection of r chosen objects. Since we already know that , Pr = n! (n - r)!' it follows that n Cr = (n - r)br!' (c) The binomial expansion states: ( stu )" = E(7). Explain why (don't demonstrate how!) this is true, by responding to the following scaffolded prompts: Imagine expanding (distributing out) the expression (2 + y)", which is copies of the factor (r + y). Multiplying all factors, (x + y)(x +y) .. . (x + y), would result in a sum of several a-degree polynomial terms that are each comprised of strings of factors of I and/or y. For example, the expansion of (2 + y)* would contain the terms (among several others) yury, yoyy, ryry, and yyyy. Since n-degree polynomial terms that consist of the same number of I factors and the same number of y factors are . (i.e., "like terms" ), they may be collected. For example, THIS = Very = TIVy = yyI = ryry = yryz and thus Byrx + yoxy + cryy + ryyz + ryry + yaya = Girly' = ()x 3y'. Similarly, whyI = yyry = yoyy = myvy and thus yyyr + yyry + yryy + ryyy = 4xly' = ()at By]. TheBinomial Theorem simply tells us how to collect and count the numbers of equivalent polynomial terms in the expansion. Why is the combination (") the appropriate way to count these? It's exactly like the Goofy Probabilist flipping her coin a times, and counting the number of paths that take her to equivalent locations! Here, the (") equivalent polynomial terms correspond to the different paths that take the Goofy Probabilist to the same location. Just as there were () ways to get to the cafe, there are () algebraically equivalent strings of a and/or y that are equal to