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COUNTING TECHNIQUES

Consider Probability Rule 6: P(E)=N(E)/N. This situation of equally likely outcomes is very common and thus computing probabilities of compound events here is just a matter of counting. There are various techniques to simplify this process of counting.

Let us define an ordered collection of k elements as a k-tuple (a pair is a 2-tuple, a triple is a 3-tuple, a quadruple is a 4-tuple etc.).

PRODUCT RULE: Consider a set of k-tuples. Suppose we have n₁ options for the first element of a tuple. After choosing the first element, we have n₂ options for the second element. After choosing the first two elements we have n₃options for the third element. And so on......after choosing the first k-1 elements we have n_k options for the k^thelement. Then the total number of k-tuples possible is equal to n₁n₂n₃....n_k-1n_k.

PERMUTATIONS: Any ordered sequence of k objects out of a total of n objects is called a permutation. The total number of such permutations possible is denoted by P_k,n = n! / (n-k)!

NOTE: Factorial of an integer m = m(m-1)(m-2)...(2)(1)

COMBINATION: Any subset (without regard to sequence or order) of k objects from a set of n objects is called a combination. The total number of such combinations possible is denoted by C_k,n = n! / (k)!(n-k)! The notation is more commonly used for C_k,n.

Permutations are larger in number than combinations: e.g., the tuples (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2) and (3,2,1) are all different permutations of the numbers 1,2 and 3. However, they all represent the same combination of numbers.

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Jayant Rajgopal