A group of four friends goes to a restaurant for dinner. The restaurant offers 12 different main dishes. (i) Suppose that the group collectively orders four different dishes to share.The waiter just needs to place all four dishes in the center of the table. How many different possible orders are there for the group? (ii) Suppose that each individual orders a main course. The waiter must remember who ordered which dish as part of the order. It’s possible for more than one person to order the same dish. How many different possible orders are there for the group? How many different passwords are there that contain only digits and lowercase letters and satisfy the given restrictions? (iii) Length is 7 and the password must contain at least one digit.
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.
A group of four friends goes to a restaurant for dinner. The restaurant offers 12 different main dishes.
(i) Suppose that the group collectively orders four different dishes to share.The waiter just needs to place all four dishes in the center of the table. How many different possible orders are there for the group?
(ii) Suppose that each individual orders a main course. The waiter must remember who ordered which dish as part of the order. It’s possible for more than one person to order the same dish. How many different possible orders are there for the group?
How many different passwords are there that contain only digits and lowercase letters and satisfy the given restrictions?
(iii) Length is 7 and the password must contain at least one digit.
(iv) Length is 7 and the password must contain at least one digit and at least one letter.
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