# Properties of Combinations

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• Properties of Combinations:
1. Consider nCn - r = nCr for 0 ≤ r ≤ n.
2. nC0 = (n!)/(0!(n - 0)!) = (n!)/(n!) = 1, because 0! = 1 as has been stated earlier.
3. If nCr = nCs, then either s = r or s = n - r.
4. "" ^nC_r = (""^nP_r)/(r!)
5. nCr + nCr - 1 = n + 1Cr
6. nC0 + nC1 + ......... nCn = 2n
7. nC0 + nC2 + nC4 + ...... = nC1 + nC3 + nC5 + ....... = 2(n - 1)
8. nCr = "" (n/r) ^(n - 1)C_(r- 1) = (n/r)((n - 1)/(r - 1)) ^(n - 2)C_(r - 2) = ....
9. nCr has maximum value if (a) r = n/2 "when n is even (b)" r = (n - 1)/2 or (n + 1)/2 when n is odd.
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