MHB Is (m²)/(m)² always an integer?

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The discussion centers on proving that the expression m²!/ (m!)² is always an integer for positive integers m. Participants express appreciation for each other's contributions and clarify details about the proof. A mistake regarding the exponent in the denominator is acknowledged and corrected, emphasizing the importance of accuracy in mathematical discussions. The overall tone is collaborative, with members actively engaging in refining the proof. This highlights the community's commitment to mathematical rigor and support.
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Prove that $\dfrac{m^2!}{(m!)^2}$ is an integer, where $m$ is a positive integer.
 
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anemone said:
Prove that $\dfrac{m^2!}{(m!)^2}$ is an integer, where $m$ is a positive integer.

Suppose a prime power $p^{2k}$ divides $(m!)^2$. It must be of the form $2k$, since $(m!)^2$ is a square. By Euclid's lemma, $p^k$ must divide $m!$, that is, $p$ divides $m!$ $k$ times. This implies that the first $k$ multiples of $p$ appear in the factorial product $m!$, since $p$ is prime. And $p \geq 2$ hence $2p \leq p^2$ and it follows that:
$$k p \leq m ~ ~ ~ \implies ~ ~ ~ 2 k p \leq k^2 p^2 \leq m^2$$
Thus the first $2k$ multiples of $p$ appear in the factorial product $(m^2)!$, and so $p^{2k}$ also divides $(m^2)!$. So the fraction is reducible to an integer in that every prime power in the denominator also appears in the numerator, QED.​
 
Bacterius said:
Suppose a prime power $p^{2k}$ divides $(m!)^2$. It must be of the form $2k$, since $(m!)^2$ is a square. By Euclid's lemma, $p^k$ must divide $m!$, that is, $p$ divides $m!$ $k$ times. This implies that the first $k$ multiples of $p$ appear in the factorial product $m!$, since $p$ is prime. And $p \geq 2$ hence $2p \leq p^2$ and it follows that:
$$k p \leq m ~ ~ ~ \implies ~ ~ ~ 2 k p \leq k^2 p^2 \leq m^2$$
Thus the first $2k$ multiples of $p$ appear in the factorial product $(m^2)!$, and so $p^{2k}$ also divides $(m^2)!$. So the fraction is reducible to an integer in that every prime power in the denominator also appears in the numerator, QED.​

Good job, Bacterius! And thanks for participating!

I will share with you and MHB the combinatorial proof of other, as in this case, I think the proof of it is pretty good and straightforward. :)

Let there be $m^2$ objects distributed in $m$ groups, each group containing $m$ identical objects. So the number of arrangement of these $m^2$ objects are $\dfrac{m^2!}{(m!)^m}$ and the the number of arrangements has to be an integer.
 
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$$\frac{(n^2)!}{(n!)^2} = \frac{(2n)! \cdot (2n + 1) \cdot (2n + 2) \, \cdots \, n^2}{(n!)^2} = \frac{(2n)!}{n! \cdot n!} \cdot k = \binom{2n}{n} \cdot k$$

As $k = (2n + 1)(2n + 2) \, \cdots \, n^2$ is an integer and binomial coefficients are all integers from their combinatorial interpretation, $(n^2)!/n!^2$ is also an integer.
 
mathbalarka said:
$$\frac{(n^2)!}{(n!)^2} = \frac{(2n)! \cdot (2n + 1) \cdot (2n + 2) \, \cdots \, n^2}{(n!)^2} = \frac{(2n)!}{n! \cdot n!} \cdot k = \binom{2n}{n} \cdot k$$

As $k = (2n + 1)(2n + 2) \, \cdots \, n^2$ is an integer and binomial coefficients are all integers from their combinatorial interpretation, $(n^2)!/n!^2$ is also an integer.

Bravo, mathbalarka!(Yes) And thanks for participating!
 
anemone said:
Good job, Bacterius! And thanks for participating!

I will share with you and MHB the combinatorial proof of other, as in this case, I think the proof of it is pretty good and straightforward. :)

Let there be $m^2$ objects distributed in $m$ groups, each group containing $m$ identical objects. So the number of arrangement of these $m^2$ objects are $\dfrac{m^2!}{(m!)^2}$ and the the number of arrangements has to be an integer.

The above is not quite correct.

it should be
So the number of arrangement of these $m^2$ objects are $\dfrac{m^2!}{(m!)^m}$

kindly note that the power is m in denominator and not 2
 
kaliprasad said:
The above is not quite correct.

it should be
So the number of arrangement of these $m^2$ objects are $\dfrac{m^2!}{(m!)^m}$

kindly note that the power is m in denominator and not 2

kaliprasad said:
the power is m in denominator and not 2

Ops...you're absolutely right. The solution I posted refers to the problem where the exponent in the denominator should be $m$. Sorry to MHB and all members and guests who have read this thread...

Thanks to you kaliprasad for pointing this out to me. I have edited my post (#3) to fix it.
 

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