Permutations and Combinations

[Jason]

\(P(n, r) = \dfrac{n!}{(n - r)!}\)

\(where\)

\(P = permutation \)

\(n = total \,\,number\,\,in\,\,set\)

\(r = number\,\, of \,\,items\,\, to \)
\(choose\,\, and \,\,arrange\)

ex. a

How many 3-note permutations can be made from a snare, bass drum, and tom note, without repetition?

\(P1: S-B-T\)

\(P2: S-T-B\)

\(P3: T-S-B\)

\(P4: T-B-S\)

\(P5: B-T-S\)

\(P6: B-S-T\)

\(P = 6\)

\(Proven\,\,via\,\,formula:\)

\(P(n, r) = \dfrac{n!}{(n - r)!}\)

\(P(3, 3) = \dfrac{3!}{(3 - 3)!} = \dfrac{3!}{(0)!} = \dfrac{6}{1} = 6\)

ex. b

What is the number of combinations in ex. a? What big idea do we see?

There is only 1 combination. This is because all those permutations are simply rearrangements of the same items.

\(Proven\,\,by\,\,formula:\)

\(C(n, r) = \dfrac{n!}{r!(n - r)!}\)

\(C(3, 3) = \dfrac{3!}{3!(3 -3)!} = \dfrac{3!}{3!(0)!} = \)
\(\dfrac{6}{6(1)} = 1\)