6. The varsity soccer team has 20 players. Three of the players are trained to be goalies while the remaining 17 can play any position. Only 11 of the players can be on the field at once. (a) We want to find the number of possible groups of 11 players the coach can choose. Is this a permutation or a combination? (b) If the coach wanted to choose her 11 starters at random by drawing names from a hat, how many possible groups of 11 starters could she choose? (c) The coach wants to make sure there is exactly one goalie on the field. How many ways can the coach choose a lineup of 11 players if exactly 1 player must be a goalie?
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
6. The varsity soccer team has 20 players. Three of the players are trained to be goalies while the
remaining 17 can play any position. Only 11 of the players can be on the field at once.
(a) We want to find the number of possible groups of 11 players the coach can choose. Is this a
permutation or a combination?
(b) If the coach wanted to choose her 11 starters at random by drawing names from a hat, how many possible groups of 11 starters could she choose?
(c) The coach wants to make sure there is exactly one goalie on the field. How many ways can the
coach choose a lineup of 11 players if exactly 1 player must be a goalie?
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