Question Number 209059 by hardmath last updated on 01/Jul/24
![Compare: 8! and 8!!](https://www.tinkutara.com/question/Q209059.png)
$$\mathrm{Compare}: \\ $$$$\mathrm{8}!\:\:\:\mathrm{and}\:\:\:\mathrm{8}!! \\ $$
Commented by mr W last updated on 01/Jul/24
![if (a/b)=1 ⇒a=b ((8!!)/(8!))=1!=1 ⇒8!!=8! ∪^(−∨∨−) _(∣<>∣)](https://www.tinkutara.com/question/Q209064.png)
$${if}\:\frac{{a}}{{b}}=\mathrm{1}\:\Rightarrow{a}={b} \\ $$$$\frac{\cancel{\mathrm{8}!}!}{\cancel{\mathrm{8}!}}=\mathrm{1}!=\mathrm{1}\:\Rightarrow\mathrm{8}!!=\mathrm{8}!\:\:\:\:\:\underset{\mid<>\mid} {\overset{−\vee\vee−} {\cup}} \\ $$
Commented by Frix last updated on 01/Jul/24
![n!!≠(n!)! n!!= { ((2^k k!, n=2k)),(((((2k)!)/(2^k k!)), n=2k−1)) :} ⇒ 8!!=(2×4)!!=2^4 ×4!=16×24=384 8!=40320 8!=105×8!!](https://www.tinkutara.com/question/Q209074.png)
$${n}!!\neq\left({n}!\right)! \\ $$$${n}!!=\begin{cases}{\mathrm{2}^{{k}} {k}!,\:{n}=\mathrm{2}{k}}\\{\frac{\left(\mathrm{2}{k}\right)!}{\mathrm{2}^{{k}} {k}!},\:{n}=\mathrm{2}{k}−\mathrm{1}}\end{cases} \\ $$$$\Rightarrow\:\mathrm{8}!!=\left(\mathrm{2}×\mathrm{4}\right)!!=\mathrm{2}^{\mathrm{4}} ×\mathrm{4}!=\mathrm{16}×\mathrm{24}=\mathrm{384} \\ $$$$\mathrm{8}!=\mathrm{40320} \\ $$$$\mathrm{8}!=\mathrm{105}×\mathrm{8}!! \\ $$
Commented by A5T last updated on 01/Jul/24
![It was probably a joke.](https://www.tinkutara.com/question/Q209087.png)
$${It}\:{was}\:{probably}\:{a}\:{joke}. \\ $$
Commented by mokys last updated on 01/Jul/24
![n! = n (n−1)(n−2)(n−3)... n!!= n (n−2)(n−4)(n−6)... 8! = 8×7×6×5×4×3×2×1 8!!= 8×6×4×2](https://www.tinkutara.com/question/Q209088.png)
$${n}!\:=\:{n}\:\left({n}−\mathrm{1}\right)\left({n}−\mathrm{2}\right)\left({n}−\mathrm{3}\right)… \\ $$$${n}!!=\:{n}\:\left({n}−\mathrm{2}\right)\left({n}−\mathrm{4}\right)\left({n}−\mathrm{6}\right)… \\ $$$$ \\ $$$$\mathrm{8}!\:=\:\mathrm{8}×\mathrm{7}×\mathrm{6}×\mathrm{5}×\mathrm{4}×\mathrm{3}×\mathrm{2}×\mathrm{1} \\ $$$$\mathrm{8}!!=\:\mathrm{8}×\mathrm{6}×\mathrm{4}×\mathrm{2} \\ $$