Factorial simplification with fractionsI am attempting to simplify the expression ( x + y 2 ) ! ( x − y 2 ) ! y ! I'm familiar with expanding expressions like y ! = ( y ) ( y − 1 ) ( y − 2 ) …but I have not encountered this before, a fraction inside a factorial. Am I looking for something like ( x + y 2 ) ! = ( x + y 2 ) ( ( x − 1 ) + ( y − 1 ) 1 ) ,and this is where I am stuck. Any help would be great.

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Factorial simplification with fractionsI am attempting to simplify the expression                    (                              x            +            y                    2                )            !              (                              x            −            y                    2                )            !              y      !      I&#039;m familiar with expanding expressions like  y  !  =  (  y  )  (  y  −  1  )  (  y  −  2  )  …but I have not encountered this before, a fraction inside a factorial. Am I looking for something like      (                  x        +        y            2        )    !  =      (                  x        +        y            2        )        (                  (        x        −        1        )        +        (        y        −        1        )            1        )    ,and this is where I am stuck. Any help would be great.

tennispopj8 · Accepted Answer

(                  x        +        y            2        )    !  =      (                  x        +        y            2        )        (                  x        +        y            2        −    1    )        (                  x        +        y            2        −    2    )    ⋯  3  ⋅  2  ⋅  1Your original expression is about as simplified as it&#039;s going to get. It is only defined if   x  ,  y are both even, or both odd.

Lucille Cummings · Accepted Answer

The expression is of the form            a      !      ⋅      b      !              (      a      −      b      )      !        ,provided   x and   y share the same parity, of course. Now this simplifies (?) to                    (                    a      b                      )              ⋅  b  !  .

Factorial simplification with fractions I am attempting to simplify the expression

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