Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, 0)=1$$

Step-by-Step Solution

Answer

For any positive integer \(n\), \(C(n, 0) = 1\).

Step 1: Understanding Combinations

To prove this, we need to understand how combinations work. The combination formula for selecting \(r\) items out of \(n\) is given by: \[ C(n, r) = \frac{n!}{r! (n-r)!} \] where \(n!\) denotes the factorial of \(n\).

Step 2: Substitute Values

Now, substitute \(r = 0\) in the combination formula. This gives us:\[ C(n, 0) = \frac{n!}{0! \cdot (n-0)!} = \frac{n!}{0! \cdot n!} \]

Step 3: Simplify the Expression

Since in mathematics, \(0!\) is defined as 1, we substitute this value into the expression:\[ C(n, 0) = \frac{n!}{1 \cdot n!} \] which simplifies to:\[ C(n, 0) = \frac{n!}{n!} = 1 \]

Step 4: Conclusion

Thus, using the definition of combinations and simplifying, we have shown that \(C(n, 0) = 1\) as required.

Key Concepts

Permutations

Permutations involve arranging a set of distinct items in a particular order. When considering permutations, the order in which we arrange items matters, making it a crucial concept in combinatorics. For instance, if you have three different books and you wish to arrange them on a shelf, each different order counts as a unique permutation.
To calculate permutations, we use the formula for the number of permutations of choosing and arranging all items from a set of size \(n\):

• \( P(n, n) = n! \)

If you're arranging only \(r\) items from a group of \(n\), the formula becomes:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

Remember, the factorial \(n!\) represents the product of all positive integers up to \(n\). Permutations are used in scenarios where the sequence of events or items is significant, such as password combinations or race placements.

Combinations

Combinations involve selecting items from a group, where the order does not matter. When we're concerned with combinations, what's important is which items are selected, not the sequence they are selected in. For example, if you need to choose 2 fruits from a basket of an apple, banana, and cherry, selecting an apple and a banana is the same as selecting a banana and an apple.
The formula for combinations is given by:

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

In this formula, \(C(n, r)\) represents the number of ways to choose \(r\) items from \(n\) available items. The presence of \(r!\) in the denominator ensures that different orders of a selection are not counted multiple times.
When considering combinations, you're often dealing with scenarios like choosing students for a team or picking lottery numbers, where the arrangement isn't important.

Factorials

A factorial, denoted by \(n!\), is a fundamental function in combinatorics that represents the product of all positive integers up to a given number \(n\). Essentially, it's a way to calculate how many ways you can arrange \(n\) items. For example, if you have 5 different marbles, \(5!\) will tell you how many different ways you can line them up.
Here are some factorial values and rules:

• \(0! = 1\) by definition, a crucial value used in key combinatorial proofs.
• \(n! = n \times (n-1) \times \dots \times 1\) for \(n > 0\)

Factorials grow rapidly with larger \(n\), which is why they are powerful in calculating permutations and combinations. Whenever you encounter a problem involving arrangements or selections in mathematics, factorials are likely to be involved significantly. They offer a stepping stone in understanding how different elements can be reconfigured or combined.