Exam 1 Questions with Solution - First-Year Interest Group Seminar | F A 1, Exams of Art

Download Exam 1 Questions with Solution - First-Year Interest Group Seminar | F A 1 and more Exams Art in PDF only on Docsity! Examination 1 Solutions CS 336 1. [5] Given sets and A B , each of cardinalityn , how many functions map in a one- to-one fashion onto ≥ 1 A B ? Let and 1 2{ , ,..., }nA a a a= 1 1: ontof A −→B . There are options for the value of n 1)(f a and, given that, 1n − options for the value of 2( )f a , …., and one option for the value of ( )nf a . Since B also has cardinality this function is automatically onto. Thus, there are such one-to-one and onto functions. n !n 2 a. [5] Given the set of symbols { , , how many different strings of length exist (allowing repetitions)? r ,..., }a a ar1 2 n ≥ 1 For each of positions there are options. Thus, there are such strings. n r nr b. [10] Given the set of r symbols , how many different strings of length exist that contain at least one and at least one ? (Assume .) { , ,..., }a a ar1 2 12n ≥ a 2a 2r ≥ There are ( strings that avoid the element a and the same number that avoid . There are subsets that avoid both elements so there are strings that either avoid a or avoid . Considering the complement, there are r r strings that contain at least one a and at least one a . 1)nr − (r − ( 2)n r − 2 1 n 2a 2( 2)n 1)r − − n 1 2)− 2a 2( 1) (n n r− − + 1 3. [10] Present a combinatorial argument that for all positive integers : n 0 3 2 n n k k n k=   =     ∑ . Consider as a model strings of length using the characters from the set { , . For each positions there are 3 options so there are 3 such strings. Alternatively, let k represent the number of positions in the string not occupied by (i.e., thus, occupied by either b or ). The value of k can vary between 0 and . For a fixed number of b s and s, there are n n k      2 , }a b c k n k n a nc c n k       c  ways to determine the positions to be occupied by the s and s and then choices (either b or c ) for each of these positions, for a total of possibilities. The remaining b 2k n k− positions must be occupied by s. Summing over all possible values of . We have such strings and this must equal . a k ≥ 0 2 n k k n k=       ∑ 3n   =    B∪ ∪ 3 n , n 3 3 n     , , 1 2,a a ,..., n r n + b. [10] Present a combinatorial argument that for all integersn : 3 33 3 3 2 3 3 2 n n n n n   + ⋅ +        (Hint: Consider three pairwise disjoint sets of cardinality .) n Let and C be pairwise disjoint sets of cardinality . Consider as a model the number of subsets of of cardinality 3. Since the cardinality of is 3 , there are  such subsets of cardinality 3. Now consider that all three elements could come from the same set or C , that two could come from one and one comes from another, and that each of the three could come from a different set. In the first case, there are 3 options for the set and then , ,A B B∪ ∪ A C 3 n   A C n  ,A B 2 3 2 n n n   ways of selecting the subset. In the second case, there are there are 3 options for the set from which 2 elements are selected, then ways of selecting those two elements, and 2 choices for the set from which only one element is selected, and finally options for that selection. In the final case, there are possible selections from each of the three sets. The total is 3 3 2 n      n n  + ⋅     +    and this must equal    . 3 3 n    4. [10] A multiset is similar to a set in that order is irrelevant but multiple copies of elements are allowed. For example, the sets {1 and { are identical and each has cardinality three but the multisets {1 and { are different and the first has cardinality three but the second has cardinality six. How many multisets of cardinality are there that employ elements from ? 2,3} 2,3} ,..., 1,1,1,2,2,3} 1,1,1,2,2,3} n ra Let us label r bins and consider the number of ways of placing n indistinguishable balls into the bins. The placements of balls into bins is in one-to- one correspondence with multisets of cardinality n that employ elements from . There are 1 2, ra a a r 1 1 2, ,..., ra a a −  such placements of balls in bins so there is the same number of multisets.     5 a. [5] How many strings are there of length using elements from the set { if repetition is not allowed. (Assume 1k ≥ 1,2,..., }n k n≤ ).