Fifty people purchase raffle tickets. Three winning tickets...

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is $\$ 500,$ in how many different ways can the prizes be awarded? How many different four-letter passwords can be formed from the letters $A, B, C, D, E, F,$ and $G$ if no repetition of letters is allowed?

Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is $\$ 500,$ in how many different ways can the prizes be awarded?
How many different four-letter passwords can be formed from the letters $A, B, C, D, E, F,$ and $G$ if no repetition of letters is allowed?

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Key Concepts

Permutations

This concept deals with arranging items in a specific order, where the sequence of the items matters. Permutations are used when the order of selection is significant, and they are calculated using factorial notation, which accounts for the number of different ways to order a subset of items taken from a larger set. This concept is essential for problems such as forming passwords or sequences, where each distinct order of elements represents a different outcome.

Combinations

This concept involves selecting items from a group where the order of selection is not important. In combinatorics, combinations are used when determining the number of ways to choose a subset of items from a larger set, without regard to the sequence in which they are chosen. The standard formula is given by the binomial coefficient, often referred to as 'n choose k.' This is applicable in situations like awarding prizes where the prize order does not affect the outcome if each prize is identical.

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