Question

10.
Find n if:
(2n)! !
7!(2n-7)! 4!(n-4)! - 24:1​

Answer 1

Answer:

14n-49+4n-16=24:1

18n-65 =24:1

18n-65/1 =24/1

18n-65=24

18n=24+65

18n=89

n=89/18

Answer 2

Answer:

n = 5

Step-by-step explanation:

²ⁿC₇ : ⁿC₄ :: 24 : 1

(2n)!/(7!)(2n-7)!  :  n!/(4!(n-4)!)  ::  24 : 1

(2n)!/(7!)(2n-7)!   = (2n)(2n-1)(2n-2)(2n-3)(2n-4)(2n-5)(2n-6) / 7!

= 2* 2 * 2 * 2 * n(n-1)(n-2)(n-3)(2n-1)(2n-3)(2n-5) /7!

= 16 * n(n-1)(n-2)(n-3)(2n-1)(2n-3)(2n-5) /5040

= n(n-1)(n-2)(n-3)(2n-1)(2n-3)(2n-5)/315

n!/(4!(n-4)!)  = n(n-1)(n-2)(n-3) / 4!

= n(n-1)(n-2)(n-3) /24

(n(n-1)(n-2)(n-3)(2n-1)(2n-3)(2n-5)/315) /(n(n-1)(n-2)(n-3) /24)  = 24/1

=> (2n-1)(2n-3)(2n-5) = 315

   9 *7 * 5 = 315

2n-1 = 9  => n = 5

Verification :

10!/(7!)(3!)  = 120

5!/4!1!  = 5

120/5 = 24