If `.^56 P_(r+6):^(54)P_(r+3)=30800 :1,` find`r`

Explanation

We have
` ""^(56)P_(r+6)=(56!)/([56-(r+6)]!)=(56!)/((50-r)!)`.
And `""^(54)P_(r+3)=(54!)/([54-(r+3)]!)=(54!)/((51-r)!)`.
`therefore (""^(56)P_(r+6))/(""^(54)P_(r+3))=(56!)/((50-r)!) xx((51-r)!)/(54!)`
`=(56xx55xx(54!))/((50-r)!) xx ((51-r)*[(50-r)!])/(54!)`
`=56xx55xx(51-r).`
Thus, `(""^(56)P_(r+6))/(""^(54)P_(r+3))=(30800)/(1)rArr 56xx55xx(51-r)=30800`
`rArr (51-r)=10rArr r=41.`
Hence, `r=41.`