PRACTICE PROBLEMS for COUNTING RULES - B Business Statistics 271 In how many ways can six people be selected from among thirteen people (a) if order counts? (b) if order does not count? 1. In how many ways can ten items be selected from among fifteen items (a) if order is important? (b) if order is not important? 2. A club has 14 members. It plans to elect four officers – a president, a vice president, a secretary and a treasurer – by secret ballot. All 14 members are eligible and willing to serve. How many possible sets of four members can serve if you ignore the office held? How many sets can be formed if the office held is considered and each member can only hold one office? 3. In the Computer Company's computer operations center, there are eight operators who must sit at one of eight individual machines that are placed one behind the other in a straight row. How many different ways could the eight operators be assigned to the eight machines? 4. The CEO of a company that produces seven different types of soup has a can of each type displayed in a row, on a credenza, in her office. (a) In how many different ways can she display the seven cans? (b) Suppose she wants to display only five of the cans at one time on the credenza. How many distinguishable arrangements are possible? 5. Two work teams, with six people on each team, are to be selected from a group of seventeen workers, with no one person serving on both teams at the same time. In how 6. many ways can these teams be chosen? Betty, Mark, and Jane belong to a club of eighteen people. A committee of twelve is to be selected at random from the membership. How many different arrangements of committee members are possible, given these facts? How many of these possible committees will definitely contain Betty, Mark, and Jane at the same time? 7.

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Permutations and combinations:

When one has to choose among a set of objects or people, then combination is used to find the number of ways. The order in which they are chosen does not matter here.

On the other hand, if one has to find the number of ways of arranging certain objects, then arrangement is used. Here, the order always matter and in such case the permutation is used.

For example, find the number of ways in which 4 benches can be occupied by 4 students. Here, first bench is occupied by 4 students, second one with the remaining 3 students, third one with remaining 2 students and 4th bench with the last student. Thus, is the case of permutation as here the order matters.

Now, suppose there are 5 seats and the number of ways of choosing 4 seats for 4 students can be chosen by using combinations. Here, the order does not matter because it does not matter which student is occupying which seat.