What's the logic behind that?

Logic Level 3

$\begin{aligned}7\forall 6&=&7\\ 4\forall 2&=&12\\ 6\forall 3&=&120\\ 8\forall 6&=&?\end{aligned}$

Define a binary operation $\forall$ such that the conditions above are fulfilled. Find function of this binary operation. And evaluate $? +8^2$ .

8 2 160 20 120 1200

1 solution

Parth Lohomi
May 4, 2015

We find that the logic is that

$x\forall y=\dfrac{x!} {y!}$

$\therefore$ $8\forall 6=\dfrac{8!}{6!}=56$ , $? =56$

$\implies$ $? +64=56+64=\boxed{120}$

Moderator note:

Good, can you convert the binary operation into a permutation function: $^a P_b$ ? Why or why not?

[Response to Challenge Master Note]

$x\forall y=\frac{x!}{y!}=\frac{x!}{(x-x+y)!}=\frac{x!}{\bigg(x-(x-y)\bigg)!}=~^x\mathrm{P}_{x-y}$

If we look at it algebraically, the conversion is valid (well-defined) for all reals $x,y\gt (-1)$ by virtue of the famous analytical extension of the factorial function, which is Gamma function.

But if we look at it combinatorially, this conversion would be valid when the permutation function has a combinatorial meaning which happens iff $0\leq y\leq x$ with the additional condition of $x,y\in\Bbb Z$ .

Prasun Biswas - 5 years, 11 months ago

What's the logic behind that?

1 solution

Moderator note:

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