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NCERT Class XI Mathematics - Permutations and Combinations - Solutions

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Question : 17
Total: 41
Find r if
(i)
5
‌
Pr = 2
6
‌
Pr−1 (ii)
5
‌
Pr =
6
‌
Pr−1
Solution:  
(i) We have,
5
‌
Pr = 2
6
‌
Pr−1
⇒
5!
(5−r)!
 = 2 ×
6!
(6−r+1)!
 ⇒
5!
(5−r)!
 =
2.6.5!
(7−r)!
⇒ 12 (5 – r) ! = (7 – r)! ⇒ 12 (5 – r)! = (7 – r) (6 – r) (5 – r)!
⇒ r2 – 13r + 30 = 0 ⇒ (r – 3) (r – 10) = 0
⇒ r = 3, 10 But r ≠ 10 (Since r ≤ 5)
∴ r = 3.
(ii)
5
‌
Pr =
6
‌
Pr−1
⇒
5!
(5−r)!
 =
6!
(7−r)!
 ⇒
1
(5−r)!
 =
6
(7−r)!
⇒ (7 – r) (6 – r) (5 – r)! = 6(5 – r)! ⇒ r2 – 13r + 36 = 0 ⇒ (r – 4)(r – 9) = 0
⇒ r = 4, 9 but r ≠ 9 (Since r ≤ 5)
∴ r = 4.
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