**Chapter 3 Permutations and Combination Ex 3.2**

## Chapter 3 Permutations and Combination Ex 3.2

**Question 1.Evaluate:(i) 8!Solution:**
8!

= 8 Ã— 7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1

= 40320

**(ii) 10!Solution:10!**

= 10 Ã— 9 Ã— 8 Ã— 7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1

= 3628800

**(iii) 10! â€“ 6!Solution:**

10! â€“ 6!

= 10 Ã— 9 Ã— 8 Ã— 7 Ã— 6! â€“ 6!

= 6! (10 Ã— 9 Ã— 8 Ã— 7 â€“ 1)

= 6! (5040 â€“ 1)

= 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1 Ã— 5039

= 3628080

**(iv) (10 â€“ 6)!Solution:**

(10 â€“ 6)!

= 4!

= 4 Ã— 3 Ã— 2 Ã— 1

= 24

**Question 2.Compute:**

= 12 Ã— 11 Ã— 10 Ã— 9 Ã— 8 Ã— 7

= 665280

= 2!

= 2 Ã— 1

= 2

**(iii) (3 Ã— 2)!Solution:**

(3 Ã— 2)!

= 6!

= 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1

= 720

**(iv) 3! Ã— 2!Solution:**

3! Ã— 2!

= 3 Ã— 2 Ã— 1 Ã— 2 Ã— 1

= 12

**Question 3.Write in terms of factorials(i) 5 Ã— 6 Ã— 7 Ã— 8 Ã— 9 Ã— 10Solution:**

5 Ã— 6 Ã— 7 Ã— 8 Ã— 9 Ã— 10 = 10 Ã— 9 Ã— 8 Ã— 7 Ã— 6 Ã— 5

Multiplying and dividing by 4!, we get

**(ii) 3 Ã— 6 Ã— 9 Ã— 12 Ã— 15Solution:**
3 Ã— 6 Ã— 9 Ã— 12 Ã— 15

= 3 Ã— (3 Ã— 2) Ã— (3 Ã— 3) Ã— (3 Ã— 4) Ã— (3 Ã— 5)

= () (5 Ã— 4 Ã— 3 Ã— 2 Ã— 1)

= (5!)

**(iii) 6 Ã— 7 Ã— 8 Ã— 9Solution:**
6 Ã— 7 Ã— 8 Ã— 9 = 9 Ã— 8 Ã— 7 Ã— 6

Multiplying and dividing by 5!, we get

**(iv) 5 Ã— 10 Ã— 15 Ã— 20Solution:**

5 Ã— 10 Ã— 15 Ã— 20

= (5 Ã— 1) Ã— (5 Ã— 2) Ã— (5 Ã— 3) Ã— (5 Ã— 4)

= () (4 Ã— 3 Ã— 2 Ã— 1)

= () (4!)

**Question 4.(i) n = 8, r = 6(ii) n = 12, r = 12(iii) n = 15, r = 10(iv) n = 15, r = 8Solution:**

**Question 5.Find n, ifSolution:**

Solution:

Solution:

**(iv) (n + 1)! = 42 Ã— (n -1)!Solution:**
(n + 1)! = 42(n â€“ 1)!

âˆ´ (n + 1) n (n â€“ 1)! = 42(n â€“ 1)!

âˆ´ + n = 42

âˆ´ + n â€“ 42 = 0

âˆ´ (n + 7)(n â€“ 6) = 0

âˆ´ n = -7 or n = 6

But n â‰ -7 as n âˆˆ N

âˆ´ n = 6

**(v) (n + 3)! = 110 Ã— (n + 1)!Solution:**

(n + 3)! = (110) (n + 1)!

âˆ´ (n + 3)(n + 2)(n + 1)! = 110(n + 1)!

âˆ´ (n + 3) (n + 2) = (11) (10)

Comparing on both sides, we get

n + 3 = 11

âˆ´ n = 8

**Question 6.Find n, if:Solution:**

âˆ´ (17 â€“ n) (16 â€“ n) (15 â€“ n) = 6 Ã— 5 Ã— 4

Comparing on both sides, we get

17 â€“ n = 6

âˆ´ n = 11

Solution:

âˆ´ 12 = (n â€“ 3)(n â€“ 4)

(n â€“ 3)(n â€“ 4) = 4 Ã— 3

Comparing on both sides, we get

n â€“ 3 = 4

âˆ´ n = 7

Solution:

âˆ´ 120 = (n â€“ 3)(n â€“ 4) (n â€“ 5)(n â€“ 6)

âˆ´ (n â€“ 3)(n â€“ 4) (n â€“ 5)(n â€“ 6) = 5 Ã— 4 Ã— 3 Ã— 2

Comparing on both sides, we get

n â€“ 3 = 5

âˆ´ n = 8

Solution:

âˆ´ (2n â€“ 1)(2n â€“ 3)(2n â€“ 5) = 9 Ã— 7 Ã— 5

Comparing on both sides. We get

âˆ´ 2n â€“ 1 = 9

âˆ´ n = 5

**Question 7.Solution:**

**Question 8.Solution:**

**Question 9.Solution:**

**Question 10.SimplifySolution:**

**Solution:**

Solution:

**(iv) n[n! + (n â€“ 1)!] + (n â€“ 1)! + (n + 1)!Solution:**

Solution:

Solution:

**Solution:**

Solution: