In a combination problem, how can the relationship between r and n be expressed?

In combination problems, the relationship between ( r ) (the number of items selected) and ( n ) (the total number of items available) is expressed through a factorial expression. The formula for combinations is given by:

[

C(n, r) = \frac{n!}{r!(n-r)!}

]

In this formula, ( n! ) (n factorial) represents the product of all positive integers up to ( n ). Similarly, ( r! ) and ( (n - r)! ) represent the factorials of ( r ) and ( n - r ), respectively. This factorial expression reflects how combinations account for the selection of items without regard to the order in which they are selected.

This approach allows for the counting of selections where the order does not matter, and the combinations can effectively be calculated using the factorial values involved in the expression. Thus, the relationship between ( r ) and ( n ) is fundamentally based on this factorial concept, making the factorial expression the correct answer.