gnatopoditw

2022-06-25

Factorial simplification with fractions

I am attempting to simplify the expression

$\frac{\left(\frac{x+y}{2}\right)!\left(\frac{x-y}{2}\right)!}{y!}$

I'm familiar with expanding expressions like

$y!=(y)(y-1)(y-2)\dots $

but I have not encountered this before, a fraction inside a factorial. Am I looking for something like

$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)\left(\frac{(x-1)+(y-1)}{1}\right),$

and this is where I am stuck. Any help would be great.

I am attempting to simplify the expression

$\frac{\left(\frac{x+y}{2}\right)!\left(\frac{x-y}{2}\right)!}{y!}$

I'm familiar with expanding expressions like

$y!=(y)(y-1)(y-2)\dots $

but I have not encountered this before, a fraction inside a factorial. Am I looking for something like

$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)\left(\frac{(x-1)+(y-1)}{1}\right),$

and this is where I am stuck. Any help would be great.

tennispopj8

Beginner2022-06-26Added 20 answers

$\left(\frac{x+y}{2}\right)!=\left(\frac{x+y}{2}\right)(\frac{x+y}{2}-1)(\frac{x+y}{2}-2)\cdots 3\cdot 2\cdot 1$

Your original expression is about as simplified as it's going to get. It is only defined if $x,y$ are both even, or both odd.

Your original expression is about as simplified as it's going to get. It is only defined if $x,y$ are both even, or both odd.

Lucille Cummings

Beginner2022-06-27Added 5 answers

The expression is of the form

$\frac{a!\cdot b!}{(a-b)!},$

provided $x$ and $y$ share the same parity, of course. Now this simplifies (?) to

$(}\genfrac{}{}{0ex}{}{a}{b}{\textstyle )}\cdot b!.$

$\frac{a!\cdot b!}{(a-b)!},$

provided $x$ and $y$ share the same parity, of course. Now this simplifies (?) to

$(}\genfrac{}{}{0ex}{}{a}{b}{\textstyle )}\cdot b!.$

Which expression has both 8 and n as factors???

One number is 2 more than 3 times another. Their sum is 22. Find the numbers

8, 14

5, 17

2, 20

4, 18

10, 12Perform the indicated operation and simplify the result. Leave your answer in factored form

$\left[\frac{(4x-8)}{(-3x)}\right].\left[\frac{12}{(12-6x)}\right]$ An ordered pair set is referred to as a ___?

Please, can u convert 3.16 (6 repeating) to fraction.

Write an algebraic expression for the statement '6 less than the quotient of x divided by 3 equals 2'.

A) $6-\frac{x}{3}=2$

B) $\frac{x}{3}-6=2$

C) 3x−6=2

D) $\frac{3}{x}-6=2$Find: $2.48\xf74$.

Multiplication $999\times 999$ equals.

Solve: (128÷32)÷(−4)=

A) -1

B) 2

C) -4

D) -3What is $0.78888.....$ converted into a fraction? $\left(0.7\overline{8}\right)$

The mixed fraction representation of 7/3 is...

How to write the algebraic expression given: the quotient of 5 plus d and 12 minus w?

Express 200+30+5+4100+71000 as a decimal number and find its hundredths digit.

A)235.47,7

B)235.047,4

C)235.47,4

D)234.057,7Find four equivalent fractions of the given fraction:$\frac{6}{12}$

How to find the greatest common factor of $80{x}^{3},30y{x}^{2}$?