A president, vice president, treasurer, and secretary, all different, are to be chosen from a club consisting of 12 people. How many different choices of officers are possible if (a) there are no restrictions? (b) A and B will not serve together? (c) C and D will serve together or not at all? (d) E must be an officer? (e) F will serve only if he is either the president or vice president?
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.
Question: A president, vice president, treasurer, and secretary, all different, are to be chosen from a club consisting of 12 people. How many different choices of officers are possible if
(a) there are no restrictions?
(b) A and B will not serve together?
(c) C and D will serve together or not at all?
(d) E must be an officer?
(e) F will serve only if he is either the president or vice president?
(As per bartleby guideline for more than 3 subparts asked, only 3 will be solved, Please upload another subparts separately.)
As given that a club consisting of 12 people.
a). If there are no restrictions:
President can be chosen in 12 ways. Then vice president can be chosen in 11 ways. the for treasure 10 ways and for secretary 9 ways.
Then the number of ways are:
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