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

Explanation

We have, `(.^(56)P_(r+6))/(.^(54)P_(r+3))=(30800)/(1)`
`implies(56)(55)(.^(54)P_(r+4))/(.^(54)P_(r+3))=(30800)/(1)` [from note (iii)]
`implies(.^(54)P_(r+4))/(.^(54)P_(r+3))=10`
`implies54-(r+4)+1=10` . . . . [from vote (v)]
`r=41`
`therefore.^(r)P_(2)=.^(41)P_(2)=41*40=1640`