Permutation, Combination, and Probability
Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written
P(n, r) = n!/(n-r)! = n(n - 1)(n - 2) ××× (n - r + 1)
Example The permutations of a, b, c two at a time are ab, ac, ba, ca, cb, and bc. The formula is P(3,2) = 3!/1! = 6.
The permutations of a, b, c three at a time are abc, bac, cab, acb, bca, and cba. Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/[r!(n - r)!].
Example The combinations of a, b, c taken 2 at a time are ab, ac, bc; taken 3 at a time is abc. An important relation is r! C(n, r) = P(n, r).
If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).
Example Two dice may be thrown in 36 separate ways. What is the probability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.
P(n, r) = n!/(n-r)! = n(n - 1)(n - 2) ××× (n - r + 1)
Example The permutations of a, b, c two at a time are ab, ac, ba, ca, cb, and bc. The formula is P(3,2) = 3!/1! = 6.
The permutations of a, b, c three at a time are abc, bac, cab, acb, bca, and cba. Each separate selection of objects that is possible irrespective of the order in which they are arranged is called a combination. The number of combinations of n things taken r at a time, written C(n, r) = n!/[r!(n - r)!].
Example The combinations of a, b, c taken 2 at a time are ab, ac, bc; taken 3 at a time is abc. An important relation is r! C(n, r) = P(n, r).
If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q).
Example Two dice may be thrown in 36 separate ways. What is the probability of throwing such that their sum is 7? Seven may arise in 6 ways: 1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1. The probability of shooting 7 is j.
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