Page 13 - 11-Math-7 Permutation Combination and Probability

71.. PQeurmadurtaattiiconECqoumatbioinnastion and Probability eeLLeeaarrnn..PPuunnjjaabb

⇒ ncr ×=r ( n n! )! ∴=n cr n!
-r
r!(n - r )!

Which completes the proof.
Corollary:

i) If r = n, then =ncn n!( n! r=)! =n! 1
n- n!0!

ii) If r = 0, then =nc0 0!( n! 0=)! =n! 1
n- 0!n!

7.3.1 Complementary Combination

Prove that: nCr = Cn
n-r

Proof: If from n different objects, we select r objects then (n - r) objects are left.

Corresponding to every combination of r objects, there is a combination of (n - r)

objects and vice versa.

Thus the number of combinations of n objects taken r at a time is equal to the number

of combinations of n objects taken (n - r) at a time.

∴ nCr =nCn-r

Other wise: cn = n!
n-r
(n - r)!(n - n + r )!

= (=n -nr!)!r! n!

r!(n - r )!

⇒ cn =ncr
n-r

Note: This result will be found useful in evaluating nCr whnecnr r> n.
2

e.g Cl2 = C12 = 12 c2 = (12).(11) = (6). (11) =66
10 12-10 2

version: 1.1

13