Question

FATORIAL E PERMUTAÇÃO SIMPLES

1. Calcule o valor dos nº fatoriais:
a) 7! =
b) 4! =
c) 2! + 3! =
d) 5! - 3! =
e) 4! . 2! =
f) 0! =


2. Quantos anagramas tem a palavra:
a)AMOR
b) LUCRO
c) TECLADO
d) TRIÂNGULO


3. Da palavra LIVRO:
a) quantos são os anagramas que começam com consoante?

b) quantos são os anagramas que começam com vogal?

Answer 1

$\boxed{\begin{array}{l}\sf Seja~m~um~n\acute umero~inteiro~n\tilde ao~negativo\,(m\in\mathbb{N}).\\\sf Conceituamos~fatorial~de~m\,(e~indicamos~por~m!)\\\sf por~meio~da~relac_{\!\!,}\tilde ao~\\\sf m!=\begin{cases}\sf m\cdot(m-1)\cdot (m-2)\cdot...3\cdot2\cdot1,para~m\geqslant2\\\sf1!=1~e~0!=1,se~m<2\end{cases}\end{array}}$

$\boxed{\begin{array}{l}\rm 1)\\\tt a)~\sf7!=7\cdot6\cdot5\cdot4\cdot3\cdot2\cdot1=5040\\\tt b)~\sf4!=4\cdot3\cdot2\cdot1=24\\\tt c)~\sf2!+3!=2\cdot1+3\cdot2\cdot1=2+6=8\\\tt d)~\sf5!-3!=5\cdot4!-3!=5\cdot24-6\\\sf=120-6=114\\\tt e)~\sf4!\cdot2!=24\cdot2=48\\\tt f)~\sf 0!=1 \end{array}}$

$\boxed{\begin{array}{l}\rm 2)\\\tt a)~\sf\underbrace{\sf AMOR}_{\sf P_4}=4!=24\\\tt b)~\sf\underbrace{\sf LUCRO}_{P_5}=5!=120\\\tt c)~\sf\underbrace{\sf TECLADO}_{P_7}=7!=5040\\\tt d)~\sf\underbrace{\sf TRI\hat ANGULO}_{P_9}=9!=362880\end{array}}$

$\boxed{\begin{array}{l}\rm 3)\\\tt a)~\sf\underbrace{\sf \boxed{\sf L}IVRO}_{3\cdot P_4}=3\cdot4!=3\cdot24=72~anagramas\\\tt b)~\sf\underbrace{\sf\boxed{\sf I}LVRO}_{2\cdot P_4}=2\cdot4!=2\cdot24=48~anagramas\end{array}}$

$\boxed{\begin{array}{l}\rm 4)~\\\tt a)~\sf P_8=8!=40320\\\tt b)~\sf P_7- P_5=7!-5!=4920\\\tt c)~\sf P_3\cdot P_6=3!\cdot 6!=4320\\\tt d)~\sf P_4+P_5=4!+5!=144\end{array}}$