Previous Exercise Permutation and Combination – Exercise 4.3
Permutation and Combination – Exercise 4.4
- Evaluate:
(i) 12P4
(ii) 75P2
(iii) 8P8
(iv) 15P1
(v) 38P0
(i) If nP4 = 20 nP2 find n
(ii)If 5Pr = 2. 6Pr-1 find r
- If nP4 : nP5 = 1:2 find n
- If 9P5 + 5. 9P4 = 10Pr , find r
Permutation and Combination – Exercise 4.4 – Solutions
- Evaluate:
(i) 12P4
(ii) 75P2
(iii) 8P8
(iv) 15P1
(v) 38P0
Solution:
(i) 12P4
nPr = n!/(n-r)!
= 12!/(12 – 4)!
= 12!/8!
= 8! 9x10x11x12/8!
= 11880
(ii) 75P2
nPr = n!/(n-r)!
= 75!/(75-2)!
= 75!/73!
= 73! 74 x 75/73!
= 5550
(iii) 8P8
nPr = n!/(n-r)!
= 8!/(8-8)!
= 8!/0!
= 8!/1
= 8!
= 40320
(iv) 15P1
nPr = n!/(n-r)!
= 15!/(15-1)!
= 14!/14! x 15
= 1/15
= 0.0667
(v) 38P0
nPr = n!/(n-r)!
= 38!/(38 – 0)!
= 38!/38!
= 1
2. (i) If nP4 = 20 nP2 find n
(ii)If 5Pr = 2. 6Pr-1 find r
Solution:
(i) nP4 = 20 nP2
We know, nPr = n!/(n-r)!
n!/(n-4)! = 20 x n!/(n-2)!
(n-4)!(n-3)(n-2)(n-1)n/(n-4)! = 20 x (n-2)! (n – 1)n/(n-2)!
(n-3)(n-2)(n-1)n = 20 x (n – 1)n
(n -3)(n – 2) = 20
n2 – 2n – 3n + 6 = 20
n2 – 5n + 6 – 20 = 0
n2 – 5n – 14 = 0
n2 – 7n + 2n – 14 = 0
n(n – 7) +2(n – 7) = 0
(n + 2)(n – 7) = 0
n = – 2 or n = 7
Therefore, in nP4 = 20 nP2 , we have n = 7
(ii)If 5Pr = 2. 6Pr-1 , we have to find r
5Pr = 2. 6Pr-1
5!/(5-r)! = 2 . 6!/(6-(r-1))!
5!/(5-r)! = 2 . 6!/(6-r+1)!
5!/(5-r)! = 2. 5! x 6/(7-r)!
1/(5-r)! = 2. 6/(5-r)!(6-r)(7-r)
1 = 2. 6/(6-r)(7-r)
1 = 12/(6-r)(7-r)
(6 – r)(7 – r) = 12
42 – 6r – 7r + r2 = 12
r2 – 13r + 30 = 0
r2 – 10r – 3r + 30 = 0
r(r – 10) -3(r – 10) = 0
(r – 10) (r – 3) = 0
⇒ r = 10 or r = 3
- If nP4 : nP5 = 1:2 find n
Solution:
nP4 : nP5 = 1:2
nP4 / nP5 = 1/2
- nP4 = nP5
- n!/(n-4)! = n!/(n-5)!
- n!/(n-5)!(n-4) = n!/(n-5)!
2 x 1/(n – 4) = 1
2 = (n – 4)
n – 4 = 2
n = 2 + 4 = 6
- If 9P5 + 5. 9P4 = 10Pr , find r
Solution:
9!/(9-5)! + 5. 9!/(9-4)! = 10!/(10 – r)!
9!/4! + 5. 9!/4!x5 = 10!/(10 – r)!
9!(1/4! + 1/4!) = 9! x 10/(10 – r)!
(1/4! + 1/4!) = 10/(10 – r)!
- 1/4! = 10. 1/(10 – r)!
1/4! = 5. 1/(10 – r)!
1/5! = 1/(10 – r)!
5! = (10 – r)!
5 = 10 – r
r = 5
Next exercise – Permutation and Combination – Exercise 4.5