Properties of C(n,r)

Properties of ⁿC_r or C(n,r)

1. ⁿC_r = ⁿC_n-r

Note: If x=y = n

ⁿC_x = ⁿC_y

2. Let n and r be non-negative integers such that r≤n. Then

ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

3. Let n and r be non-negative integers such that 1≤ r≤n. Then
ⁿC_r = (n/r) ^n-1C_r-1

4. If 1≤ r≤n, then

n.^n-1C_r-1 = (n-r+1)ⁿC_r-1

5. ⁿC_x = ⁿC_y implies x+y = n

6. If n is even, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_n/2.

7. If n is odd, then the greatest value of ⁿC_r [0≤ r≤n] is ⁿC_(n+1)/2 or ⁿC_(n-1)/2.