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Chapter 6: Problem 78

The formula for \(C(n, r)\) is written as a ratio of two whole numbers. Can \(C(n, r)\) ever be a fraction and not a whole number? Explain (without actually discussing the formula itself).

Short Answer

Expert verified
\(C(n, r)\), the binomial coefficient, represents the number of ways to choose r elements from a set of n elements without considering order. As it counts combinations, it must always be a whole number, as you cannot have fractional combinations. Therefore, \(C(n, r)\) can never be a fraction and will always be a whole number.

Step by step solution

01

Know what C(n, r) represents

C(n, r) is also known as the binomial coefficient, which is the number of ways to choose r elements from a set of n elements, without regard to the order of the elements. In other words, it represents the count of different combinations of n elements taken r at a time.
02

Consider the nature of combinations

Since C(n, r) represents the count of possible combinations, it must result in a whole number, as you cannot have a fraction of a combination. The number of ways to choose elements from a set can only be a whole number.
03

Provide examples

For instance, let's say we have a set with four elements {A, B, C, D}, and we want to find the number of ways to choose two elements from this set (C(4,2)). The possible combinations are {A, B}, {A, C}, {A, D}, {B, C}, {B, D}, and {C, D}. As we can see, there are 6 possible combinations, which is a whole number.
04

Conclusion

Since C(n, r) represents the number of ways to choose r elements from a set of n elements without considering order, it will always result in whole numbers, as you cannot have fractional combinations. Therefore, C(n, r) can never be a fraction and will always be a whole number.

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