The number of possible hands where order does matter

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The number of possible hands where order does matter is 52 P 5 52 51 50 49 48. But we want to only count different hands, not different orderings. Given any 5 cards, they can be arranged in 5 4 3 2 1 5! ways. So the number of different hands is 52 51 50 49 48 5! 2,598,960. 95
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Generally, when we select k objects from n where we disregard order, we say we have a combination of length k . Sometimes this is written as n C k . The general formula is n C k n P k k ! n ! k ! n k ! , which is often written as n k n ! k ! n k ! and read as “ n choose k .” It is the number of combinations of n (distinct) objects chosen k at a time. 96
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In the example of dealing a 5-card hand from a deck of 52 cards, the number of possible hands is 52 5  , which we already calculated. Remember, this is without regard to order. 97
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EXAMPLE : Consider the lottery example, where 6 different digits from 0 to 9 must be chosen but where one wins by matching the 6 digits regardless of the order chosen. So, this is an unorderd setup where selection is without replacement. We have 10 objects to choose from – the digits from 0 to 9 – and we are selecting 6 of them. Therefore, the number of possible combinations is 10 6 10! 6!4! 10 9 8 7 4 3 2 210 When order does not matter, the chance of winning is 1/210 compared with 1/ 10!/4! 1/151,200 if we have to get the order right. 98
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We have shown how to fill in three of the four entries in the table, where we have n objects and select k if them: Without Replacement With Replacement Ordered n ! n k ! n k Unordered n k n k 1 k See Casella and Berger for the lower-right-hand entry. 99
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Often we need to apply the multiplication rule with the counting rules. EXAMPLE : Suppose a soccer team consists of 18 players: 4 forwards, 6 midfielders, 6 backers, and 2 goalies. The coach wants to field a team with 2 forwards, 4 midfielders, 4 backers, and (by rule) one goalie. How many different combinations of players are possible, assuming that, within a position, each player can play anywhere on the field (for example, a forward can play left forward or right forward)? 100
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SOLUTION: The total number is obtained by multiplying the number of possible lineups within each position. So think of choosing each position as a “task”: 4 2 6 4 6 4 2 1 6 15 15 2 2,700. 101
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7 . 2 . Computing Probabilities Using Counting : Combinatorics Counting is often used to obtain probabilities of complicated events. With an equally likely model, probabilities are easy to obtain once we find the number of elements in the even of interest and the number of outcomes in the sample space: P A #ofelementsin A We sometimes have to combine counting with the other rules for computing probabilities.
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The number of possible hands where order does matter is 52...

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