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2723Execution Schedules With at Most One Large SliceAn execution schedule reaches total size 11 using slices of sizes 1, 2, and 5, but the large 5-lot slice may be used at most once. Order of slices does not matter. How many schedules are possible?脑筋急转弯中等derivation未尝试面试订阅2724Recurrence Hidden in a Rational Generating FunctionLet A(x)=sum n>=0 a n x n = (1+x)/(1-2x+x 3). Derive a linear recurrence for a n and compute a 5.脑筋急转弯困难derivation未尝试面试订阅2725Coefficient in a Mixed Pole Generating FunctionFind the coefficient of x 8 in 1/((1-x) 2(1-x 3)).脑筋急转弯中等derivation未尝试面试订阅2726Subset Count With Cardinality Divisible by ThreeHow many subsets of a 10-element set have size divisible by 3? Solve it with a roots-of-unity filter or an equivalent generating-function argument.脑筋急转弯中等derivation未尝试面试订阅2727Binary Strings With Ones Congruent to One Mod FourHow many binary strings of length 9 have a number of ones congruent to 1 modulo 4?脑筋急转弯简单数值题未尝试面试订阅2728Ternary Strings With a Modular Count of One SymbolHow many strings of length 8 over the alphabet A,B,C contain a number of As congruent to 2 modulo 3?脑筋急转弯中等derivation未尝试面试订阅2729Chip Allocation With a Congruence ConstraintTwelve identical chips are allocated across four desks. How many allocations have the property that the combined chips on desks 1 and 2 are divisible by 3?脑筋急转弯困难derivation未尝试面试订阅2730Submultisets With Size Divisible by ThreeFrom the multiset a,a,a,b,b,c,c,c,c , you may choose any submultiset. How many choices have total size divisible by 3?脑筋急转弯困难derivation未尝试面试订阅2731Coefficient in a Product of Short Generating FunctionsFind the coefficient of x 12 in (1+x+x 2) 4 (1+x 2+x 4) 2.脑筋急转弯中等derivation未尝试面试订阅2732Weak Compositions With Uniform Upper BoundsHow many weak compositions of 15 into 5 parts have every part at most 4? Solve it from the generating function ((1-x 5)/(1-x)) 5.脑筋急转弯中等derivation未尝试面试订阅2733Payment Combinations With an Odd Number of Nine-Unit NotesA cashier can use 2-unit, 5-unit, and 9-unit notes to make total payment 24. The number of 9-unit notes must be odd. Order does not matter. How many valid combinations exist?脑筋急转弯中等derivation未尝试面试订阅2734Weighted Sum With One Binary Heavy BlockHow many nonnegative integer solutions satisfy 2a+3b+4c+6d=20 if d is restricted to be either 0 or 1?脑筋急转弯困难derivation未尝试面试订阅2735Compositions Into Ones and TwosLet c n be the number of ordered compositions of n using only parts 1 and 2. Use a generating function to show c n = F n+1 , and then compute c 10.脑筋急转弯简单数值题未尝试面试订阅2736A Fibonacci Binomial Sum From Coefficient ExtractionEvaluate sum k>=0 C(12-k, k). You should explain why a generating function turns this into a Fibonacci-type coefficient.脑筋急转弯中等derivation未尝试面试订阅2737Inventory Paths With Holds InsertedA trader executes 4 buys, 4 sells, and 2 holds. Buys and sells each change inventory by plus or minus 1, holds leave inventory unchanged. The trader starts flat, ends flat, and inventory may never become negative at any intermediate time. How many execution sequences are possible?脑筋急转弯困难derivation未尝试面试订阅2738Flat Inventory Paths With Five Buys and Five SellsA flat-start inventory process consists of exactly 5 buys and 5 sells, each changing inventory by plus or minus 1. Inventory may never go negative. How many feasible paths are there?脑筋急转弯中等derivation未尝试面试订阅2739Catalan Generating Function for Balanced Trade SkeletonsLet C n count buy/sell skeletons of length 2n that start and end at inventory 0 and never go negative. Write the generating function C(x)=sum n>=0 C n x n, derive the quadratic equation it satisfies, and compute C 4.脑筋急转弯困难derivation未尝试面试订阅2740Coefficient in 1/((1-x^2)^2(1-x^3))Find the coefficient of x 10 in 1/((1-x 2) 2(1-x 3)).脑筋急转弯中等derivation未尝试面试订阅2816PGF of a Binomial VariableLet X\sim Binomial (n,p). Derive its probability generating function G X(s) and use it to recover E[X].概率中等derivation未尝试面试订阅2817Sum of Independent Poisson CountsLet X\sim Poisson (\lambda 1) and Y\sim Poisson (\lambda 2) be independent. Use PGFs to identify the distribution of X+Y.概率中等derivation未尝试面试订阅