Permutations with repetition - practice problems
Permutations with repetition count arrangements where some objects are identical, reducing the total number of distinct permutations. When arranging n objects where n₁ are of one type, n₂ of another, and so on, the formula is n!/(n₁!n₂!...nₖ!). For example, the word "MISSISSIPPI" has 11 letters with repetitions, giving 11!/(4!4!2!) distinct arrangements. This differs from standard permutations where all objects are unique. When unlimited repetition is allowed (like selecting with replacement), k items chosen from n options give n^k permutations. Applications include counting distinct arrangements of letters in words, distributing identical objects, and analyzing sequences with repeated elements. Understanding this concept is essential for advanced combinatorics and probability calculations.Directions: Solve each problem carefully. Show all your work.
Number of problems found: 39
- Kenneth 2
Kenneth has 100 pennies, 20 nickels, 10 dimes, and 4 quarters. How many ways can he choose coins that total 25 cents? - Morse code 2
We have two characters, a dot and a comma. How many two-element and how many three-element characters can be created with repetition? - Beads
We have 4 beads. One is green, one is yellow, and 2 are pink. In how many possible ways can we string them on a string? - Grouping - combinatorics
In how many different ways can 24 people be divided into: a) 6 groups of the same size. b) Groups of 5, 6, 7, and 6 people. c) Groups of 4, 5, 7, and 8 people. - Numbers 6D
Find out how many natural six-digit numbers exist whose digit sum is four. - Refrigerator, lemonades
How many possible ways can we store three lemonades, four mineral waters, and two juices in the refrigerator next to each other? - Probability - shelf
Ten books are placed randomly on one shelf. Find the probability that certain three books are placed next to each other. - Play match
A hockey match played for three periods ended with a score of 2:3. How many possibilities are there on how the given thirds could have been completed? - Monogram letter combinations
Calculate how many different monograms (short name and surname) I can make from the letters A, E, M, Z, and K. a) with repetition: b) without repetition: - Aquaristics
We consider “words” (i.e. arbitrary strings of letters) obtained by rearranging the letters of the word “AQUARISTICS”. All letters are distinguishable from each other here. The number of such words that also contain the expression “CAVA” (as consecutive l - Dice sum probability What is the probability that a roll of three dice will result in a number less than 7?
- Positive integer integral How many different sets of a positive integer in the form (x, y, z) satisfy the equation xyz=1400?
- Apple pear division
How many ways can you divide two identical apples and: a) 3, b) 4, c) 5 identical pears between Janka and Mařenka? - Wagon assembly ways
How many ways can we assemble five wagons when sand is in three wagons and cement in two? - Word letter creation
How many different four-letter words can we create from the letters of the word JAMA? - Word OPTICAL
Find the number of possible different arrangements of the letters of the word OPTICAL such that the vowels would always be together. - Four-digit number digits
Please find out how many different four-digit numbers we can create from the digits 3 and 8 so that the two digits three and two digits eight are used in each four-digit number created. - Dice snake colors
Jeníček has 4 identical yellow dice and 3 identical blue dice. How many different colored snakes can make them? - Comma dot characters
How many characters can we create from two commas and four dots? - Boy girl placement
Three boys and four girls. How many ways can they be placed side by side according to gender?
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