In this exercise, give an expression for the answer using...

In this exercise, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate. Baskin-Robbins Ice Cream. Burt Baskin and Irv Robbins began making ice cream in $1945 .$ Initially they developed 31 flavors - one for each day of the month. (Source: Baskin-Robbins) a) How many 2 -dip cones are possible using the 31 original flavors if order of flavors is to be considered and no flavor is repeated? b) How many 2 -dip cones are possible if order is to be considered and a flavor can be repeated? c) How many 2 -dip cones are possible if order is not considered and no flavor is repeated?

In this exercise, give an expression for the answer using permutation notation, combination notation, factorial notation, or other operations. Then evaluate.
Baskin-Robbins Ice Cream. Burt Baskin and Irv Robbins began making ice cream in $1945 .$ Initially they developed 31 flavors - one for each day of the month. (Source: Baskin-Robbins)
a) How many 2 -dip cones are possible using the 31 original flavors if order of flavors is to be considered and no flavor is repeated?
b) How many 2 -dip cones are possible if order is to be considered and a flavor can be repeated?
c) How many 2 -dip cones are possible if order is not considered and no flavor is repeated?

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Key Concepts

Permutation without Repetition

This concept is used when order matters and no item is repeated. In such cases, the first selection is made from the full set of options, and each subsequent selection is made from a reduced set because no repeats are allowed. The total number of outcomes is the product of the available options for each position, often written using permutation notation such as P(n, r), where n represents the total items and r is the number of positions selected.

Permutation with Repetition

This counting principle applies when order matters and items can be chosen more than once. Each selection is made from the full set regardless of previous choices, meaning that the total number of outcomes is the number of choices raised to the power of the number of selections, expressed as n^r. This is useful when repetitions are allowed in the ordering.

Combination without Repetition

This concept is used when the order of selection does not matter and no items are repeated. The focus is on choosing a subset of items from a larger set, and duplicate sets that differ only by the arrangement of items are considered the same. The number of outcomes is given by the binomial coefficient, commonly written as C(n, r) or 'n choose r', which counts the distinct groups possible.