Permutations and combinations help!!!

If nC6 = nC4, find the value of n. the clue is (n-4)(n-5)=30. but i do not know how to get that.

PLEASE HELP!
THANKS!

nC6 = nC4

=> n! / (n-6)!6! = n! / (n-4)!4!

=> (n-6)!6! = (n-4)!4! [ after eliminating same numerator from both side]

=> (n-6)! . 6.5.4! = (n-4).(n-5).(n-6)! . 4! [as we know n! = n. (n-1) . (n-2) . . . . 4.3.2.1]

=> 6.5 = (n-4)(n-5)

=> n^2 -9n +20 = 30

=> n^2 -9n -10=0

=> n^2-n-10n-10=0 [spilliting middle term]

=> n(n-1) -10(n-1) =0

=> (n-1)(n-10)=0

this impiles n=1,10.
n cant be 1 becoz factorial cant be negative and thus n=10. your required answer.

hope you get it!!!!

n! / 6!(n-6)! = n! / 4!(n-4)!

Divide both sides by n! and take the reciprocal

6!(n-6)! = 4!(n-4)!
6*5*4! (n-6)! = 4!(n-4)! [because 6! = 6*5*4!]

6*5*4! (n-6)! = 4! (n-4)(n-5)(n-6)! [because (n-4)! = (n-4)(n-5)(n-6)!]

Divide both sides by (n-6)!4!

6*5 = (n-4)(n-5)
30 = (n-4)(n-5)
30 = n^2 -9n +20
0 = n^2 -9n -10
0 = (n-10) (n+1)

therefore n = 10, n = -1

But -1 doesn't make mathematical sense, so n is just 10.

You can substitute those values into 10C6 and 10C4 to test it (210).

nC6=n!/(6!(n-6)!)
simalar to nC4 = n!/(4!(n-4)!)
equate
n!cancels
4!(n-4)!=6!(n-6)!
remember that
n!=n(n-1)(n-2)...
4!(n-4)(n-5)(n-6)!=6!(n-6)!
cancel
4!=24
6!=720
24(n-4)(n-5)=720
(n-4)(n-5)=30
n^2-9n-10
n -10
n 1
n=10
n=-1
n>4>0 because nCr n>r
n=10

nC6 = nC4
OR nC6 = nCn--4
whence 6 = n -- 4 OR n = 10