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Find Wrd{3,{a, b, c, d, e}, OM, RA} and its cardinality.

Find Wrd{3,{a, b, c, d, e}, OM, RA} and its cardinality.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I am having really hard time solving this question. please help me. I have attached the lecture slides too. I hope it helps you to solve this problem. I am stuck. Please help.

Rmk: This leads to four different kinds of word selections: Topic 2.3.1: Counting Selections Wrd{n, E, OM, RA}: This is an n-letter selection where the order matters and repetition is allowed. This is the least restricted type of selection (and as we shall see therefore the most numerous). A word selection from this set is called an n-tuple (or an n-string). Def: Suppose n, k E N and ECU is an alphabet where v(E) = n. We define the cardinalities of the four word selections as follows: Wrd{n. E.OMKÁl: This is an n-letter selection where order matters and repetition is not permitted. A word selection from this set is called an n-permutation. T(n, k) := v(Wrd{k,E,0M,RA}) (k-tuples of n options) P(n, k) := v(Wrd{k, E, OM,RA}) (k-permutations of n options) Wrd{n, E.OM.RA}: This is an n-letter selection where order does not matter (that is only the number of letters is recorded, not the order in which they were selected) and repetition is permitted. M(n, k) := v(Wrd{k,E,OM,RA}) (k-multicombinations of n options) C(n, k) := v(Wrd{k,E, QM,RAÍ) (k-combinations of n options) A word selection from this set is called an n-multicombination. T(4,2) = 16 P(4,2) = 12 Ex: Wrd{n, E. OM RAÁ: This is an n-letter selection where neither order matters nor is repetition permitted. This is the most restrictive case (and therefore the least numerous). A word selection from this set is called an n-combination. M(4,2) = 10 C(4, 2) = 6 Ex: Suppose E= {a, b, c, d}. Then Topic 2.3.1: Word Sets Wrd{2,E,OM, RA} = {aa, ab, ac, ad, ba, bb, bc, bd, ca,cb,cc, cd, da, db, dc, dd} Suppose E is a non-empty set of elements called "letters" and n E N. We will call E an alphabet. A selection of letters from E is called a word. If a word has letters, we say it has length n and is an n-letter word. Wrd{2,E,OM RA} = {ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc} Def: Wrd{2, E,OM, RA} = {aa, ab,ac, ad, bb, bc, bd, cc, cd, dd} If the letters are selected one-at-a-time and the ordered is recorded, we say that order matters (OM). If the letters are chosen all-at-once or the order is not recorded, we say order does not matter (OM). Def: Wrd{2, E, QM,BA) = {ab, ac, ad, bc, bd, cd} Facts: For all n e N, E E U: If a letter may be selected more than once, we say repetition is allowed (RA), If letters may not be selected more than once, then repetition is not allowed (RA). Def: 1) Wrd{n, E,OM RA} C Wrd{n, E,QM, RA} C Wrd{n, E, OM, RA} Spec: Suppose n eN and E is an alphabet. We specify all possible n-letter word selections (from E) using the notation: 2) Wrd{n, E, QM, BA} S Wrd{n, E, OM,BAJ C Wrd{n, E, OM,RA} Wrd{n, E, OM?,RA? } 3) Wrdfn, ΣρM, RAJ n Wrd in, Σ, OM,.R:- Wrdfn, Σ,ρΜ ) where OM? And RA? would be replaced with OM or OM and RA or RA respectively. 4) Wrd{n, E, OM, RA} U Wrd{n,E, OM,BAJ = Wrd{n,E, OM, RA} Ex: 1) 6! = (6)(5)(4)(3)(2)(1) = 720. 2) 103 = (10)(9)(8) = 720. 3) 103 = (10)(11)(12) = 1320. Facts: For n EN 1) n! = n = 1 2) 0! = nº = n°l = 1 3) n = n =n 4) nk=n! (n-k); provided 0sksn. 5) nk = (n+k=1)! (n-1)! provided n > 0.
Transcribed Image Text:Rmk: This leads to four different kinds of word selections: Topic 2.3.1: Counting Selections Wrd{n, E, OM, RA}: This is an n-letter selection where the order matters and repetition is allowed. This is the least restricted type of selection (and as we shall see therefore the most numerous). A word selection from this set is called an n-tuple (or an n-string). Def: Suppose n, k E N and ECU is an alphabet where v(E) = n. We define the cardinalities of the four word selections as follows: Wrd{n. E.OMKÁl: This is an n-letter selection where order matters and repetition is not permitted. A word selection from this set is called an n-permutation. T(n, k) := v(Wrd{k,E,0M,RA}) (k-tuples of n options) P(n, k) := v(Wrd{k, E, OM,RA}) (k-permutations of n options) Wrd{n, E.OM.RA}: This is an n-letter selection where order does not matter (that is only the number of letters is recorded, not the order in which they were selected) and repetition is permitted. M(n, k) := v(Wrd{k,E,OM,RA}) (k-multicombinations of n options) C(n, k) := v(Wrd{k,E, QM,RAÍ) (k-combinations of n options) A word selection from this set is called an n-multicombination. T(4,2) = 16 P(4,2) = 12 Ex: Wrd{n, E. OM RAÁ: This is an n-letter selection where neither order matters nor is repetition permitted. This is the most restrictive case (and therefore the least numerous). A word selection from this set is called an n-combination. M(4,2) = 10 C(4, 2) = 6 Ex: Suppose E= {a, b, c, d}. Then Topic 2.3.1: Word Sets Wrd{2,E,OM, RA} = {aa, ab, ac, ad, ba, bb, bc, bd, ca,cb,cc, cd, da, db, dc, dd} Suppose E is a non-empty set of elements called "letters" and n E N. We will call E an alphabet. A selection of letters from E is called a word. If a word has letters, we say it has length n and is an n-letter word. Wrd{2,E,OM RA} = {ab, ac, ad, ba, bc, bd, ca, cb, cd, da, db, dc} Def: Wrd{2, E,OM, RA} = {aa, ab,ac, ad, bb, bc, bd, cc, cd, dd} If the letters are selected one-at-a-time and the ordered is recorded, we say that order matters (OM). If the letters are chosen all-at-once or the order is not recorded, we say order does not matter (OM). Def: Wrd{2, E, QM,BA) = {ab, ac, ad, bc, bd, cd} Facts: For all n e N, E E U: If a letter may be selected more than once, we say repetition is allowed (RA), If letters may not be selected more than once, then repetition is not allowed (RA). Def: 1) Wrd{n, E,OM RA} C Wrd{n, E,QM, RA} C Wrd{n, E, OM, RA} Spec: Suppose n eN and E is an alphabet. We specify all possible n-letter word selections (from E) using the notation: 2) Wrd{n, E, QM, BA} S Wrd{n, E, OM,BAJ C Wrd{n, E, OM,RA} Wrd{n, E, OM?,RA? } 3) Wrdfn, ΣρM, RAJ n Wrd in, Σ, OM,.R:- Wrdfn, Σ,ρΜ ) where OM? And RA? would be replaced with OM or OM and RA or RA respectively. 4) Wrd{n, E, OM, RA} U Wrd{n,E, OM,BAJ = Wrd{n,E, OM, RA} Ex: 1) 6! = (6)(5)(4)(3)(2)(1) = 720. 2) 103 = (10)(9)(8) = 720. 3) 103 = (10)(11)(12) = 1320. Facts: For n EN 1) n! = n = 1 2) 0! = nº = n°l = 1 3) n = n =n 4) nk=n! (n-k); provided 0sksn. 5) nk = (n+k=1)! (n-1)! provided n > 0.
Find Wrd{3,{a, b, c, d, e},OM, RA} and its cardinality.
Transcribed Image Text:Find Wrd{3,{a, b, c, d, e},OM, RA} and its cardinality.
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