Many states in U.S.A have a lottery game, usually called a Pick-4, in which you pick a fourdigit number such as 7359. During the lottery drawing, there are four bins, each containing balls numbered 0 through 9. One ball is drawn from each bin to form the four-digit winning number. There are many variations of this game. The primary variation allows you to win if the four digits in your number are selected in any order as long as they are the same four digits as obtained by the lottery agency. For example, if you pick four digits making the number 1265, then you will win if 1265, 2615, 5216, 6521, and so forth, are drawn. The variations of the lottery game depend on how many unique digits are in your number. Consider the following four different versions of this game. Find the probability that you will win this lottery in each of these situations. i. All four digits are unique (e.g., 1234) ii. Exactly one of the digits appears twice (e.g., 1223 or 9095) iii. Two digits each appear twice (e.g., 2121 or 5588)
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.
Many states in U.S.A have a lottery game, usually called a Pick-4, in which you pick a fourdigit number such as 7359. During the lottery drawing, there are four bins, each containing balls
numbered 0 through 9. One ball is drawn from each bin to form the four-digit winning number. There are many variations of this game. The primary variation allows you to win if the four
digits in your number are selected in any order as long as they are the same four digits as
obtained by the lottery agency. For example, if you pick four digits making the number
1265, then you will win if 1265, 2615, 5216, 6521, and so forth, are drawn. The variations
of the lottery game depend on how many unique digits are in your number. Consider the
following four different versions of this game. Find the probability that you will win this
lottery in each of these situations.
i. All four digits are unique (e.g., 1234)
ii. Exactly one of the digits appears twice (e.g., 1223 or 9095)
iii. Two digits each appear twice (e.g., 2121 or 5588)
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