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Summary of permutations (arrangements where the order counts)

Summary of permutations (arrangements where the order counts). r -permutation from n different objects without repetition: r -permutation from n different objects with repetition:. - permutations of n different objects with limited repetition.

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Summary of permutations (arrangements where the order counts)

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  1. Summary of permutations (arrangements where the order counts) • r-permutation from n different objects without repetition: • r-permutation from n different objects with repetition:

  2. - permutations of n different objects • with limited repetition How many numbers from 1, 1, 1, 2, 2, 3 can be constructed? Ans:

  3. Combinations (selections without reference to the order) • r-combination from n different objects • Example: 3-combinations from {a, b, c, d} abc acd abd bcd acd adc adb bdc bcd dca bad cdb bdc dac bda cbd cab cad dab dcb cba cda dba dbc {a,b,c} {a,c,d} {a,b,d} {b,c,d}

  4. r-combinations of n objects without repetition The equivalence of 3-combinations from 4 objects and permutations of 4 objects with 3 of the same type {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}

  5. Combinations with repetitions. Take, for instance, 4-combinations of {a,b}: {a, a, a, a}, {a, a, a, b}, {a, a, b, b}, {a, b, b, b}, {b, b, b, b} We can consider this problem as the arrangements of 4 identical objects and one separator |: {a, a, a, a} ****| {a, a, a, b} ***|* {a, a, b, b} **|** {a, b, b, b} *|*** {b, b, b, b} |**** 5-permutations of 5 objects if 4 of the them are identical:

  6. Combinations with repetitions. Donut shop has 5 types of donuts. In how many ways we can select ten donuts? This problem can be represented as an equivalent arrangement of ten donuts into 5 boxes. All possible “distributions” Can be considered as “permutations” of a dozen of donuts and 4 separators between boxes: One possible arrangement:

  7. 14! = C (14, 4) 10!4! We need to count the number of permutations of 10 donuts and 4 separators. So, we have 14 objects, 4 of which are identical and 10 are identical. From another side, any arrangement can be viewed asa selection of 4 numbers out of 14 (or 10 out of 14) 1 2 3 4 5 6 7 8 9 1011 12 13 14

  8. The number of r-combinations of n objects that can be repeated (any number of times) Can be considered as the number of arrangements of r identical objects and n-1 separators (bars).

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