Question

3. Prove that cnr=cnn-r​

Answer 1

To prove ---> ⁿcᵣ = ⁿcₙ_ᵣ

Proof---> We have a formula from permutation and combination

n! = 1×2×3.........................×n

n! , means number of ways to arrange n things

We have one more formula

(1) ⁿcᵣ = n! / r! × (n-r)!

Now ,

RHS = ⁿcₙ_ᵣ

= n! / (n-r)! × { n- ( n - r ) !}

= n! / (n-r)! × (n - n + r )!

= n! / (n - r )! r !

= n! / r! × ( n - r )!

= ⁿcᵣ = RHS

Additional information--->

(1) ⁿpᵣ = n! / ( n - r )!

(2) (n + 1 )! = ( n+1 ) × ( n!)

(3) ⁿcᵣ + ⁿcᵣ_₁ = ⁿ⁺¹cᵣ

(4) n × ⁿ⁻¹cᵣ_₁ = ( n - r + 1 ) × ⁿcᵣ_₁