r ज्ञात कीजिए , यदि
(i) ` .^(5)P_(r) = 2 .^(6)P_(r-1)" "(ii) .^(5)P_(r) = .^(6)P_(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)!) = (2 xx 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 cancelgt n "अतः " 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)!) = (6xx 5!)/((7-r)(6-r)(5-r)!)`
` rArr 1 = 6/((7-r)(6-r))`
` rArr (7-r)(6-r)= 6 "या " 42 - 13r + r^(2) = 6`
` rArr r^(2) - 13r + 36 = 0 "या " (r-4)(r-9) = 0`
` rArr r = 9,4 ` परन्तु ` r cancelgt n` अतः `r ne 9`
` rArr r = 4`