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Exploring Permutations, Combinations, and Repeated Selections in Mathematics

Learn about permutations, combinations, and selections including repetitions, with examples like constructing numbers and selecting donuts efficiently. Understand the strategies behind various arrangements of objects and how to calculate them accurately.

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Exploring Permutations, Combinations, and Repeated Selections in Mathematics

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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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