Find r if (i) `^5P_r=2^6P_(r-1)` (ii) `^5P_r=^6P_(r-1)`

Explanation

(i) `.^(5)P_(r) = 2.^(6)P_(r-1)`
`rArr (5!)/((5-r)!) = 2 xx (6!)/((6-r+1)!)`
`rArr (5!)/((5-r)!) = 2 xx (6!)/((7-r)!)`
`rArr ((7-r)!)/((5-r)!) = (2xx 6!)/(5!)`
`rArr ((7-r)(6-r)(5-r)!)/((5-r)!) = (2xx 6 xx 5!)/(5!)`
`rArr (7-r) (6-r) = 12`
`rArr 42 - 13r +r^(2) = 12`
`rArr r^(2)-13r +30 = 0`
`rArr (r-10) (r-3) = 0`
`rArr r = 3, 10 ( :' r cancel(gt) n` so `r ne 10)`
`:. r = 3`
(ii) `.^(5)P_(r) = .^(6)P_(r-1)`
`(5!)/((5-r)!) = (6!)/((6-r+1)!)`
`rArr (5!)/((5-r)!) = (6!)/((7-r)!)`
`rArr (5!)/((5-r)!) = (6 xx 5!)/((7-r)(6-r)(5-r)!)`
`rArr 1 = (6)/((7-r)(6-r))`
`rArr (7-r) (6-r) = 6` or `42 - 13r +r^(2) = 6`
`rArr r^(2) -= 13r +36 = 0` or `(r-4) (r-9) = 0`
`rArr r = 9,4` but `r cancel(gt) n` so `r ne 9`
`rArr r - 4`