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By Marcel B. Finan

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Solution. We will use T for tail and H for head. (a) The sample space is composed of eight simple events: S = {T T T, T T H, T HT, T HH, HT T, HT H, HHT, HHH}. (b) The event of obtaining more than one head is the set {T HH, HT H, HHT, HHH} Probability is the measure of occurrence of an event. Various probability concepts exist nowadays. A widely used probability concept is the experimental probability which uses the relative frequency of an event and is defined as follows. Let n(E) denote the number of times in the first n repetitions of the experiment that the event E occurs.

A) How many different three-number combinations can be made? (b) How many different combinations are there if the three numbers are different? 5 (a) Miss Murphy wants to seat 12 of her students in a row for a class picture. How many different seating arrangements are there? (b) Seven of Miss Murphy’s students are girls and 5 are boys. In how many different ways can she seat the 7 girls together on the left, and then the 5 boys together on the right? 6 Using the digits 1, 3, 5, 7, and 9, with no repetitions of the digits, how many (a) one-digit numbers can be made?

N! = 1. (n−1)! = n and C(n, n − 1) = (n−1)! = n. (n−n)! n! n! (n−n+k)! (n−k)! = C(n, k). (c) We have n! n! (n − k + 1)! (n − k)! (n − k + 1)! (n − k + 1)! n! (n − k + 1)! (n + 1)! (n + 1 − k)! 1. 8 The Chess Club has six members. In how many ways (a) can all six members line up for a picture? (b) can they choose a president and a secretary? (c) can they choose three members to attend a regional tournament with no regard to order? 42 COUNTING AND COMBINATORICS Solution. (a) P (6, 6) = 6! = 720 different ways (b) P (6, 2) = 30 ways (c) C(6, 3) = 20 different ways As an application of combination we have the following theorem which provides an expansion of (x + y)n , where n is a non-negative integer.

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