Question

2. How many positive integers less than 1000 are multiples of 5 or 7? Explain your answer using the inclusion-exclusion principle 3. For the purpose of this problem, a word is an ordered string of 5 lowercase letters from the English alphabet (i.e., the 26 letters from a to z). For example, "alpha" and "zfaxr" are words. A subword of a word is an ordered string that appears as consecutive letters anywhere within the given word. For example, "cat" is a subword of "cathy" and 'xycat" but not a subword of "caaat". (a) How many words are there? (b) How many words have no repeated letters? (c) How many words contain the letter a exctly three times? (d) How many words do not contain the subword "cat"? Determine the number of functions f: X → Y Let X = {1, 2, 3,4) and Y = {a,b,c,d,e). that satisfy the following conditions. 4, (a) No restrictions (b) f is injective (d) 1 Ef-(a)

Answer 1

Suppose , A is ser of au fos 'H几ïnle ers Less than 1000 nrtch are muui ple of5 and B is set of alu rositire indees than 10oo whiare mulHple of hen (mutti ple of 5 n(A)-1000-1 199 5 nemu-iple of 7 1000 - 28( muupl both 5 and T) So, numbes of au tosi ve inee les thaa 1000 (AUB) n(A) + n(B)-n(Ana) = 199 142 28 = 313 a) Totat owercase aphabe 26 A word is an ordered string of 5 lowercase letert from the Engläh alphabe tesers may be repeated) number of words6Xx26 2x2 26 26 26 2626 ら 11 88 1376 C5 number of words having no reheated ㅶ,Lebens leters and awarge them Select 20 26 51 21 26X25 X 24X 23X22 7493600 ニ

of- word เ contal ning Leer a exact three times Cx 25 25 Select any three lacerem five plates C) btoz numboc b to z and pu 'd Utsre In the "rect two下lacet , Þur an lekter fr m stoz 10*25米25 d) number of word Containing subwod cat са 2028 So C at numbe% of words .. .not containing subword "Cat . Total wor)-(numbor of words containing suword "Cat" 11881376- 2028 1187-9:348 1XI 4 ( .) number injec ive อช one-one function Ixl -120 (c) number of funcHors such thak fC) -a is 3 4.

(d numbes of Runcions such that ef(a) 1,2,3,4 (30+%,+3c2+ 3c3 \ -4' so, furuian: number of such S12