Combination Of N Different Objects Assignment Help

Combination of n different objects:

i) The number of all combinations of n distinct objects taken r at a time is given by;

C(n, r) = n!/r! (n – r) !

where n! is a factorial.

Example: Evaluate C(10, 2)

Solution: Here,

n = 10

And r = 2

Using the general formula

C(n, r) = n!/ r! (n – r) !

we get

C(10, 2) = 10!/ 2! (10 – 2)!

= 10 x 9 x 8!/2! 8!

= 45

ii) C(n, n) = 1

Proof: we have,

C(n, r) = n!/ r! (n – r) !

putting r = n

C(n, r) = n!/ n! (n – n) !

= n!/ n! 0!

we know that

0! = 1

therefore,

C(n, n) = 1

iii) C(n, r) = C(n, n-r) for 0≤ r ≤ n

Proof: C(n, n-r) = n!/ (n – r) ! x [n – (n – r)] !

= n!/ (n – r) ! x (r) !

= C(n, r)

iv) C(n, r) + C(n, r – 1) = C(n+1, r)

Proof:

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