Factorial Divided By Factorial And That To So Many Times

Can You Prove That,

$(nk)!$ is always divisible by ${ (n!) }^{ k }$

#Combinatorics #NumberTheory #Factorial #Divisibility

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

As a multinomial coefficient, $\frac{(a_1 + a_2 + \dots + a_k)!}{a_1! a_2! \dotsm a_k!}$ is an integer for any nonnegative integers $a_1$ , $a_2$ , $\dots$ , $a_k$ . Taking $a_i = n$ for all $i$ , you get that $\frac{(nk)!}{(n!)^k}$ is an integer.

Jon Haussmann - 7 years, 3 months ago

i was not aware about this theorem , so i had used combinatorics to prove the above question thanks for sharing

Harsh Depal - 7 years, 3 months ago

Can you share your combinatorics approach?

Calvin Lin Staff - 7 years, 2 months ago

@Calvin Lin – Yes we can prove ,

$\\ \frac { ({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+........+{ a }_{ k })! }{ { a }_{ 1 }!{ a }_{ 2 }!{ a }_{ 3 }!....{ a }_{ k }! }$

Using Combinatorics

Let us assume that we have ${ a }_{ 1 }$ number of objects of 1 type , ${ a }_{ 2 }$ number of objects of some type and ... ${ a }_{ k }$ number of objects of some other type; So We Get Total Number Of Objects As

$\\ \\ Total\quad Objects\quad =\quad ({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+....+{ a }_{ k })$

Therefore Using Combinatorics We Get Number Of Ways To Arrange The Objects As

$\\ \frac { ({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+........+{ a }_{ k })! }{ { a }_{ 1 }!{ a }_{ 2 }!{ a }_{ 3 }!....{ a }_{ k }! }$

Number Of Ways To Arrange Will Be Obviously An Integer , Hence We Have Proved That

$\\ \frac { ({ a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+........+{ a }_{ k })! }{ { a }_{ 1 }!{ a }_{ 2 }!{ a }_{ 3 }!....{ a }_{ k }! }$

Will Always Be An Integer

Harsh Depal - 7 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$