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Business Mathematics and Statistics · Ch 2 — Algebra (Partial Fractions, Permutations, Combinations, Mathematical Induction, Binomial Theorem)

Combinations — nCr and Its Properties

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Combinations — nCr and Its Properties

A combination is a selection of objects where order does not matter — choosing A then B is the same selection as choosing B then A.

The number of combinations of nn distinct things taken rr at a time, written nCr^{n}C_{r}, is

nCr=n!r! (n−r)!^{n}C_{r} = \frac{n!}{r!\,(n-r)!}

Relationship to permutations. Since a combination followed by arranging the chosen rr objects in every possible order produces every permutation of those rr objects,

nPr= nCr×r!^{n}P_{r} = \, ^{n}C_{r} \times r!

This is the key link between the two ideas: a combination counts which objects are chosen; multiplying by r!r! additionally counts the order in which they are arranged.

Key properties.

  • nCr= nCn−r^{n}C_{r} = \, ^{n}C_{n-r} — choosing rr objects to include is the same act as choosing n−rn-r objects to leave out.
  • nC0= nCn=1^{n}C_{0} = \, ^{n}C_{n} = 1.
  • nCr+ nCr−1= n+1Cr^{n}C_{r} + \, ^{n}C_{r-1} = \, ^{n+1}C_{r} (Pascal's rule) — this is exactly the rule that generates Pascal's triangle, where each entry is the sum of the two entries above it.
nnRow of nCr^{n}C_{r} values (r=0,1,2,…,nr=0,1,2,\ldots,n)
01
11, 1
21, 2, 1
31, 3, 3, 1
Definition 1nCr (Combination of n things taken r at a time)

nCr = n!/(r!(n-r)!), the number of ways to select r objects from n distinct objects when order …

Definition 2Pascal's Rule

nCr + nC(r-1) = (n+1)Cr; the rule that generates each entry of Pascal's triangle as the sum of the two entrie …

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