10th mathematics exercise questions with answers, Mathematics

Permutation and Combination – Exercise 4.4 – Class X

Ashoora Arif

Previous Exercise Permutation and Combination – Exercise 4.3 

Permutation and Combination – Exercise 4.4

  1. 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

  1. If nP4 : nP5 = 1:2 find n
  2. If 9P5 + 5. 9P4 = 10Pr­ , find r

Permutation and Combination – Exercise 4.4 – Solutions

  1. 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

  1. If nP4 : nP5 = 1:2 find n

Solution:

nP4 : nP5 = 1:2

nP4 / nP5 = 1/2

  1. nP4 = nP5
  2. n!/(n-4)! = n!/(n-5)!
  3. n!/(n-5)!(n-4) = n!/(n-5)!

2 x 1/(n – 4) = 1

2 = (n – 4)

n – 4 = 2

n = 2 + 4 = 6

  1. 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. 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

 

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Published by Ashoora Arif