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Business Mathematics and Basic Statistics · Ch 8 — Permutations and Combinations

Combinations — Meaning and Formula (nCr)

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Combinations — Meaning and Formula (nCr)

Meaning of a combination

A combination is a SELECTION of a group of objects where the order of picking does not matter. Choosing 3 students — say Asha, Bina and Chitra — for a committee is the SAME outcome no matter which order their names were called in. That is the hallmark of a combination problem: order does not matter.

The formula

Every combination of rr objects can itself be arranged in r!r! different orders, and each such arrangement is one of the outcomes nPr^{n}P_{r} counts. So the number of combinations is the number of permutations divided by r!r!:

nCr=nPrr!=n!r! (n−r)!,0≤r≤n, n,r∈W^{n}C_{r} = \frac{^{n}P_{r}}{r!} = \frac{n!}{r!\,(n-r)!}, \qquad 0 \le r \le n,\ n,r \in \mathbb{W}

Note

Two identities worth knowing

nC0=nCn=1^{n}C_{0} = {}^{n}C_{n} = 1 (there is exactly one way to choose nothing, and exactly one way to choose everything); and nCr=nCn−r^{n}C_{r} = {}^{n}C_{n-r} — choosing rr objects to INCLUDE is the same count as choosing the remaining (n−r)(n-r) objects to LEAVE OUT. Always check whether the smaller-looking nCn−r^{n}C_{n-r} makes a calculation quicker.

Evaluating nCr^{n}C_{r} …

Definition 1Combination

A combination of rr objects chosen from nn distinct objects is a SELECTION of those rr objects where order does not matter. Two selections containing the same objects, …

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