# If nC12 = nC8 then what is the value of n?please answer this with brief explination.

Vedant
35 Points
6 years ago

Dear student
The formula for ‘n choose k’ is
n!/(n-k)!k!

Therefore we have
n!/(n-8)!8! = n!/(n-12)!12!
(n-8)!8! = (n-12)!12!                 ….1

Now there are 2 methods to proceed- guess and check and algebraic
I will first present the guess and check method

To make equation 1 true we must try to equate n-8 to 12 and n-12 to 8
Now we have 2 equations n – 8 = 12 and n – 12 = 8 which give us n = 20

Now the algebraic approach

(n – 8)!8! = (n – 12)!12!
(n – 8)! = (n – 12)! x 12 x 11 x 10 x 9
(n – 8)! / (n – 12)! = 12 x 11 x 10 x 9

On replacing n – 8 by p, we get

p! / (p – 4)! = 12 x 11 x 10 x 9
p x (p-1) x (p-2) x (p-3) = 12 x 11 x 10 x 9
Therefore p = 12 and n – 8 = p
n – 8 = 12
Hence n = 20

Regards
Vedant
Soumendu Majumdar
159 Points
6 years ago
Whenever you see the expression to be
$_{r}^{n}\textrm{C} =_{s}^{n}\textrm{C}$
then either r = s   or   r + s = n
Here r = 12 and s = 8
Clearly r is not equal to s
so    n = r + s
implies      n = 12 + 8
implies      n = 20  (Ans)

Hope it helps!
P.S – This is the shortest method in which you can do these kinds of sums rather than breaking them into factorials on both sides.
ASh
62 Points
6 years ago

Whenever you see the expression to be
$_{r}^{n}\textrm{C} =_{s}^{n}\textrm{C}$
then either r = s   or   r + s = n
Here r = 12 and s = 8
Clearly r is not equal to s
so    n = r + s
implies      n = 12 + 8
implies      n = 20  (Ans)