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Exercício de De quantas maneiras podem ser escolhidos três números naturais distintos
Exercício:
De quantas maneiras podem ser escolhidos três números naturais distintos, de 1 a 30, de modo que sua soma seja par?
Ora, a soma de 3 números naturais será um número par se, e somente se, os 3 números forem pares, ou se dois forem ímpares e um for par. Nestas condições, teremos: De 1 a 30, existem 15 números pares ( e 15 números ímpares). Realmente, considerando-se a PA acima de primeiro termo 2 e último termo 30, sendo a razão igual a 2, teremos: 30 = 2 + (n – 1) .2 \ n = 15.
Olha penso eu que a resposta da maria está incompleta, pois bem, sabemos que a soma de três números inteiros será par se, e somente se, dois números forem impares ou se 1 for par, deste modo teremos
será uma terna de números (x, y, z) onde x, y, z pertence aos conjunto formado pelos números 01 ao 30, assim teremos 2 casos
se todos forem pares, então será as maneiras será o produto de 15x14x13 pois existem 15 números pares de 1 a 30 e como quero números distntos se no primeiro escolho 15 só sobrará 14 para o segundo e para o terceiro sobrará 13 assim, temos
15x14x13=2730
se 2 forem pares e 1 impar, teremos
15x15x14, pois teremos 15 pares, 15 impares e sobrando 14 impares assim temos
15x15x14 = 3150
Assim,
as maneiras serão 3150 + 2730 = 5880 maneiras.
desculpem naum entendi a resposta da maria e ado csfab está errada,pois o gabarito é 2030 maneiras mas eu naum sei como chegar a essa resposta usando apenas a análise combinatória. alguém pode me ajudar?
quatro maneiras
4 mmmmmaaaaannnneeeeiiiiirrrraaaasss
acho que 15! 30:2=15
dê 5 representação diferente do número natuaral 4.
15! = 15.14.13.12!
3! 12! 3.2.1.12!
455
total de nº pares =15!
três nº pares =3!
nº distintos = 15-3 = 12!
15! = 15.147.13! = 105 x 15 = 1.575
2! 13! 2.1.13!
1.575 + 455 = 2.030
2 nº impares = 2!
distintos 15 – 2 = 13!
15 possibilidades de nº pares no 3º algortimo.
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