`r` का मान ज्ञात कीजिए, यदि
`(i) '""^(5)P_(r)=""^(6)P_(r-1)`
`(ii) 5""^(4)P_(4)=6""^(5)P_(r-1)`

Options

A. `(i) 4 , (ii) 3`
B. `(i) 4 , (ii) 4`
C. `(i) 3 , (ii) 3`
D. `(i) 3 , (ii) 4`

Answer

A

Explanation

(i) `:' .^(5)P_(4)=.^(6)P_(r-1)`
`(5!)/((5-4)!)=(6!)/((6-r+1)1`
`(5!)/((5-r)!)=(6.5!)/((6-r+1)(6-4)(6-r-1)!)`
`1/((5-r)!)=6/((7-r)(6-r)(5-4)!)`
`(7-r)(6-r)=6`
`42-6r-7r+r^(2)=6`
`r^(2)=13r+36=0`
`(r-9)(r-4)=0`
`implies r=9` या `r=4`
अत: `r` का मान 9 सम्‍भव नहीं है।
इसलिए `r=4`
(ii) `:' 5.^(4)P_(r)k=6.^(5)P_(r-1)`
`:. 5. (4!)/((4-r)!)=6. (5!)/((5-r+1)!)`
`implies (5.4!)/((4-r)!)=(6.5!)/((5-r+1)(5-r)(5-r-1)!)`
`implies(5!)/((4-r)!)=(6.51)/((5-r+1)(5-r)(4-r)!)`
`implies (6-r)(5-r)=6`
`implies 30-5r-6r+r^(2)=6`
`impliesr^(2)-11r+24=0`
`implies (r-8)(r-3)=0`
`implies r=8` या `r=3`
`:'r` का मान 8 सम्भव नहीं है। अतः `r` का सही मान 3 होगा।