![]() Hence, there are C(n, r) bit strings of length "n" that contain exactly "r" 1s. The positions of "r" 1s in a bit string of length "n" form an r-combination of the set How many bit strings of length n contain exactly "r" 1s? Of 6-combinations of a set with 30 elements, because the order in which these people are chosen The number of ways to select a crew of six from the pool of 30 people is the number Many ways are there to select a crew of six people to go on this mission (assuming that all crew A group of 30 people have been trained as astronauts to go on the first mission to Mars. The answer is given by the number of 5-combinations of a set with 10 elements. How many ways are there to select five players from a 10-member tennis team to make a trip to Many ways are there to select 47 cards from a standard deck of 52 cards?īecause the order in which the five cards are dealt from a deck of 52 cards does not matter, there are C(52, 5) = 52! / 5!47! different hands of five cards that can be dealt. How many poker hands of five cards can be dealt from a standard deck of 52 cards? Also, how The number of r-combinations of a set with n elements, where n is a nonnegative integer and r is an integer with 0 ≤ r ≤ n, equals Of the letters ABCDEFGH in which ABC occurs as a block Number of permutations of six objects, namely, the block ABC and the individual letters D, E,īecause these six objects can occur in any order, there are 6! = 720 permutations How many permutations of the letters ABCDEFGH contain the string ABC ?īecause the letters ABC must occur as a block, we can find the answer by finding the With minimum distance, and she computes the total distance for each possible path, she must If, for instance, the saleswoman wishes to find the path between the cities Seven elements, because the first city is determined, but the remaining seven can be ordered The number of possible paths between the cities is the number of permutations of How many possible ordersĬan the saleswoman use when visiting these cities? She must begin her trip in a specifiedĬity, but she can visit the other seven cities in any order she wishes. Suppose that a saleswoman has to visit eight different cities. The number of different ways to award the medals is the number of 3-permutations Many different ways are there to award these medals, if all possible outcomes of the race can finisher receives a silver medal, and the third-place finisher receives a bronze medal. The winner receives a gold medal, the second place Suppose that there are eight runners in a race. Three prize winners is the number of ordered selections of three elements from a set of 100Įlements, that is, the number of 3-permutations of a set of 100 elements. Winner from 100 different people who have entered a contest?īecause it matters which person wins which prize, the number of ways to pick the How many ways are there to select a first-prize winner, a second-prize winner, and a third-prize If n and r are integers with 0 ≤ r ≤ n, then P(n, r) = n! / (n − r)! 1 = 120 ways to arrange all five students in a line for.The second in four ways, the third in three ways, the fourth in two ways, and the fifth in one To arrange all five students in a line for a picture, we select the first student in five ways, 3 = 60 ways to select three students from a group of five students to stand.Have been selected, there are three ways to select the third student in the line. There are four ways to select the second student in the line. ![]() To select the first student to stand at the start of the line. In how many ways can we select three students from a group of five students to stand in line forĪ picture? In how many ways can we arrange all five of these students in a line for a picture?įirst, note that the order in which we select the students matters. Thus, an r-combination is simply a subset of the set with "r" elements. R-permutations of a set with "n" distinct elements.Īn r-combination of elements of a set is an un-ordered selection of "r" elements from the set. ![]() If "n" is a positive integer and "r" is an integer with 1 ≤ r ≤ n, then there are Number of distinct elements of a set of a particular size, where the order of these elementsĬombination : finding the number of ways to selectĪ particular number of elements from a set of a particular size, where the order of the elementsĪ permutation of a set of distinct objects is an ordered arrangement of these objects. ![]() Permutation : finding the number of ways to arrange a specified ![]()
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