^{n}C

_{r}or C(n,r)

1.

^{n}C

_{r}=

^{n}C

_{n-r}

Note: If x=y = n

^{n}C

_{x}=

^{n}C

_{y}

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

^{n}C

_{r}+

^{n}C

_{r-1}=

^{n+1}C

_{r}

3. Let n and r be non-negative integers such that 1≤ r≤n. Then

^{n}C

_{r}= (n/r)

^{n-1}C

_{r-1}

4. If 1≤ r≤n, then

n.

^{n-1}C

_{r-1}= (n-r+1)

^{n}C

_{r-1}

5.

^{n}C

_{x}=

^{n}C

_{y}implies x+y = n

6. If n is even, then the greatest value of

^{n}C

_{r}[0≤ r≤n] is

^{n}C

_{n/2}.

7. If n is odd, then the greatest value of

^{n}C

_{r}[0≤ r≤n] is

^{n}C

_{(n+1)/2}or

^{n}C

_{(n-1)/2}.

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