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Handout_6 PERMUTATIONS AND COMBINATIONS

# Handout_6 PERMUTATIONS AND COMBINATIONS - PERMUTATIONS AND...

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PERMUTATIONS AND COMBINATIONS ( 03/15/09) #6 THE COUNTING (MULTIPLICATION) PRINCIPLE . If one activity can occur in any of m ways, and, following this , a second activity can occur in any of n ways, then both activities can occur in the order given in m*n ways. Example . Let us consider a bank containing 4 coins: a quarter ( Q ), a dime ( D ), a nickel ( N ), and a penny ( P ). If one coin is taken out of the bank at random (the first activity), then the set of 4 possible outcomes (sample space) is {Q, D, N, P } . If a coin is tossed (the second activity), there are 2 possible outcomes, heads ( h ) and tails ( t ), and the sample space is {h, t} . If a coin is taken out of the bank and tossed, then the sample space is the set of 4*2 = 8 possible outcomes: { (Q, h), (Q, t); (D, h), (D, t); (N, h), (N, t); (P, h), (P, t) } Problem 1 . A bank contains 4 different coins: a quarter ( Q ), a dime ( D ), a nickel ( N ), and a penny ( P ), which are to be drawn out one at a time without replacement . In how many different orders can the 4 coins be removed from the bank? Solution . There are 4 possible outcomes for the first draw: Q, D, N, P. --------------------------------------------------------- Then, since 3 coins remain in the bank, | Number of Coins | Number of | there are 3 possible coins to be drawn on Draw | in the Bank | Possible Outcomes| the 2 nd draw, and in accordance with the ---------------------------------------------------------| counting principle there are 4*3 = 12 1 st | 4 | 4 | possible orders in which the first two 2 nd | 3 | 4*3 | coins could be removed: 3 rd | 2 | 4*3*2 | (Q, D), (Q, N), (Q, P); (D, Q), (D, N), (D, P); 4 th | 1 | 4*3*2*1 | (N, Q), (N, D), (N, P);

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Handout_6 PERMUTATIONS AND COMBINATIONS - PERMUTATIONS AND...

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