# Operations with factorials

#### jones123

Hi, i'm having troubles understanding operations with faculty.

If:

(3k)! (k+1)! (2(k+1))!
__________________
k! (2k)! (3(k+1))!

Then why is this equal to:

(k+1)(2k+1)(2k+2)
________________
(3k+1)(3k+2)(3k+3)

Thanks for the help!

#### Kmath

Note that
$$\displaystyle (3(k+1))!=(3k+3)!=(3k+3)(3k+2)(3k+1)(3k)!$$.
And do it again with 2 instead of 3.

1 person

#### Soroban

MHF Hall of Honor
Hello, jones123!

I must assume that you know how to "cancel" factorials.

$$\displaystyle \text{Show that: }\: \frac{\big(3k\big)!\,\big(k+1\big)!\,\big(2[k+1]\big)!}{k!\,\big(2k\big)!\,\big(3[k+1]\big)!} \;=\;\frac{(k+1)(2k+1)(2k+2)}{(3k+1)(3k+2)(3k+3)}$$

$$\displaystyle \text{We have: }\:\frac{(3k)!\,(k+1)!\,(2k+2)!}{k!\,(2k)!\,(3k+3)!}$$

. . . . . $$\displaystyle =\;\frac{(3k)!}{(3k+3)!}\cdot \frac{(k+1)!}{k!}\cdot\frac{(2k+2)!}{(2k)!}$$

. . . . . $$\displaystyle =\;\frac{1}{(3k+3)(3k+2)(3k+1)}\cdot\frac{k+1}{1} \cdot\frac{(2k+2)(2k+1)}{1}$$

. . . . . $$\displaystyle =\;\frac{(k+1)(2k+1)(2k+2)}{(3k+1)(3k+2)(3k+3)}$$

But this reduces further . . .

. . . . . $$\displaystyle =\;\frac{(k+1)\,(2k+1)\,2(k+1)}{(3k+1)\,(3k+2)\,3(k+1)}$$

. . . . . $$\displaystyle =\;\frac{2(k+1)(2k+1)}{3(3k+1)(3k+2)}$$

1 person

Thanks!