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lect2_1 - Summary of permutations(arrangements where the...

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Summary of permutations (arrangements where the order counts) r -permutation from n different objects without repetition: r -permutation from n different objects with repetition: - = = - = 1 if , )! 1 ( 1 , 0 if , 1 ! ) ( , )! ( ! ) , ( n n n n n n r r n n r n P r R n r n P = ) , (
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! !... ! )! ... ( 2 1 2 1 n n k k k k k k + + + - permutations of n different objects with limited repetition n k k k + + + ... 2 1 How many numbers from 1, 1, 1, 2, 2, 3 can be constructed? 1 k 2 k 3 k ! 2 ! 3 ! 6 Ans:
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Combinations (selections without reference to the order) r -combination from n different objects Example : 3-combinations from {a, b, c, d} ! ) , ( ) , ( r r n C r n P = 24 ! 4 ) 3 , 4 ( = = P abc acd abd bcd acd adc adb bdc bcd dca bad cdb bdc dac bda cbd cab cad dab dcb cba cda dba dbc ! 3 ) 3 , 4 ( = C {a,b,c} {a,c,d} {a,b,d} {b,c,d}
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= - = = r n r r n n r r n P r n C ! )! ( ! ! ) , ( ) , ( r -combinations of n objects without repetition {a, b, c, d} 1 1 1 0 {a, b, c} 1 1 0 1 {a, b, d} 1 0 1 1 {a, c, d} 0 1 1 1 {b, c, d} The equivalence of 3-combinations from 4 objects and permutations of 4 objects with 3 of the same type
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