Problem 81 - 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.) $$C(n, n-1)=n$$

Step-by-Step Solution

Answer

The statement is true; \( C(n, n-1) = n \).

Step 1: Understand What C(n, n-1) Represents

The expression \( C(n, n-1) \) represents the combination of \( n \) items taken \( n-1 \) at a time. This is the number of ways to choose \( n-1 \) items from \( n \) total items without regard to the order of selection.

Step 2: Review the Combination Formula

The formula for combinations is given by:\[C(n, r) = \frac{n!}{r!(n-r)!}\]where \(n!\) denotes the factorial of \(n\).

Step 3: Substitute n and r into the Combination Formula

Using the combination formula, substitute \( r = n-1 \) and simplify:\[C(n, n-1) = \frac{n!}{(n-1)!(n-(n-1))!} = \frac{n!}{(n-1)! \, 1!}\]

Step 4: Simplify the Expression

The expression simplifies as follows:\[\frac{n!}{(n-1)! \, 1!} = \frac{n \times (n-1)!}{(n-1)! \, 1} = \frac{n \times (n-1)!}{(n-1)!} = n\]The \((n-1)!\) terms cancel out, leaving \(n\).

Step 5: Conclusion

Therefore, we have demonstrated that \( C(n, n-1) = n \). This means there are exactly \( n \) ways to choose \( n-1 \) items from \( n \) items.

Key Concepts

Factorials

Factorials are a fundamental concept in mathematics, particularly in permutations and combinations. A factorial of a non-negative integer \( n \), denoted by \( n! \), is the product of all positive integers less than or equal to \( n \). This can be defined as:

• \( n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \)
• By convention, \( 0! = 1 \)

Factorials are crucial in calculations where order and arrangement are involved.
When using factorials in combination formulas, they help to compute how different sets of elements can be arranged. Understanding how to manipulate factorials is essential, as they simplify complex problems in various mathematical computations.
The example in our problem uses factorials to compute combinations, showcasing how efficiently these elements can be chosen without regard to order.

Permutations

Permutations involve arranging a set of items in a specific order. Unlike combinations, order matters in permutations. The number of permutations of \( n \) items taken \( r \) at a time is calculated using the formula:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This formula tells us how many different sequences can be formed by taking \( r \) elements from a group of \( n \) elements.
Permutations are used when the arrangement is key, such as ordering objects or determining possible passwords.
If we think about the problem \( C(n, n-1) \), permutations indirectly play a part. While combinations don't emphasize order, the knowledge of arranging \( n-1 \) items from \( n \) is intrinsic to understanding the arrangement possibilities.

Combinatorial Proof

A combinatorial proof is a type of mathematical proof that uses counting techniques to establish the validity of a statement. It involves showing that two expressions count the same set of objects in different ways.
The problem presented uses a combinatorial proof with the concept \( C(n, n-1) = n \).

• Understand it as selecting \( n-1 \) items from \( n \).
• Each selection process inherently means leaving out exactly 1 item.

By calculating \( \frac{n!}{(n-1)! \cdot 1!} \), we see that it equals \( n \), confirming that there are \( n \) ways to choose \( n-1 \) items.
This proof not only verifies the formula but also illustrates a fundamental aspect of combinations: how counting ways in different methods leads to the same result. Combinatorial proofs are powerful because they offer deeper insights into how counting principles work, often providing more intuitive understanding of mathematical identities.