Maharashtra State BoardHSC Commerce 11th
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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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