Question

Consider the following problem: How many ways can a committee of four people be selected from a group of 10 people? There are many other similar problems in which you are asked to find the number of ways to select a set of items from a given set of items. The general problem can be stated as follows: Find the number of ways r different things can be chosen from a set of n items, in which r and n are nonnegative integers and r n. Suppose C(n, r) denotes the number of ways r different things can be chosen from a set of n items. Then, C(n, r) is given by the following formula: Cðn;rÞ ¼ n! r!ðn rÞ! in which the exclamation point denotes the factorial function. Moreover, C(n, 0) = C(n, n) = 1. It is also known that C(n, r) = C(n – 1, r – 1) + C(n – 1, r). a. Write a recursive algorithm to determine C(n, r). Identify the base case(s) and the general case(s). b. Using your recursive algorithm, determine C(5, 3) and C(9, 4).

Short answer

C(5, 3) = 10, C(9, 4) = 126.

Step-by-step solution

Understanding Factorials and Combinations

To solve this problem, we use combinations, which determine how many ways we can choose a subset from a larger set. The formula for combinations is \( C(n, r) = \frac{n!}{r!(n-r)!} \). Factorials are used here, where \( n! \) is the product of all positive integers up to \( n \). Base cases include \( C(n, 0) = C(n, n) = 1 \), as choosing all or none from a set gives only one possible selection.

Identifying Base and General Cases for Recursion

For the recursive determination of \( C(n, r) \), the base cases are: if \( r = 0 \) or \( r = n \), return 1, because there is only one way to choose none or all items from a set. The general case can be derived from the identity \( C(n, r) = C(n-1, r-1) + C(n-1, r) \).

Writing the Recursive Algorithm

The recursive algorithm in pseudo-code can be written as follows:1. If \( r = 0 \) or \( r = n \), return 1.2. Otherwise, return \( C(n-1, r-1) + C(n-1, r) \).

Calculating C(5, 3) Using Recursion

To calculate \( C(5, 3) \):- Use the identity \( C(5, 3) = C(4, 2) + C(4, 3) \).- Further, \( C(4, 2) = C(3, 1) + C(3, 2) \) and \( C(4, 3) = C(3, 2) + C(3, 3) \).- Continue breaking down: - \( C(3, 1) = C(2, 0) + C(2, 1) = 1 + 2 = 3 \) - \( C(3, 2) = C(2, 1) + C(2, 2) = 2 + 1 = 3 \) - \( C(3, 3) = 1 \)- Finally, \( C(5, 3) = 3 + 3 + 3 = 10 \).

Calculating C(9, 4) Using Recursion

To calculate \( C(9, 4) \):- Use the identity \( C(9, 4) = C(8, 3) + C(8, 4) \).- Continue recursively splitting terms till reaching base cases: - Example paths: \( C(8, 3) = C(7, 2) + C(7, 3), C(7, 2) = C(6, 1) + C(6, 2), etc. \)- Perform the recursive calculations to accumulate their values.- Final calculation yields that \( C(9, 4) = 126 \).

Key concepts

Recursion

Recursion is a fundamental concept in computer science and mathematics where a function calls itself in order to solve smaller instances of the same problem until a base case is reached. This technique is particularly useful for tasks such as calculating combinatorics, where problems can be broken down into simpler, sub-problems. Recursion involves two key parts:

• Base Case(s): This is the simplest, smallest instance of the problem, where an explicit solution is known. For instance, in our combinations problem, if you want to choose all items or none at all, there is exactly one way to do that, which makes this our base case.
• General Case(s): Here, the function reduces the problem into one or more smaller sub-problems, relying on the fact that the same solution steps apply. In combinatorics, this is represented by the identity: \(C(n, r) = C(n-1, r-1) + C(n-1, r)\). This breaks the problem into two smaller sub-problems of the same nature.

By using recursion, we can elegantly solve complex problems by systematically solving and combining results of smaller instances of the problem.

Factorials

Factorials are a key component in combinatorics and play a crucial role in calculating combinations. The factorial of a non-negative integer \( n \) is denoted by \( n! \) and is calculated as the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorials grow rapidly, even for relatively small values of \( n \).

Factorials are used in the formula for combinations:

• The number of ways to choose \( r \) items from \( n \) items is calculated as \( C(n, r) = \frac{n!}{r!(n-r)!} \).
• The terms \( r! \) and \( (n-r)! \) in the denominator adjust for the order of selection not being important in combinations.

Understanding factorials is essential for grasping how to compute combinations and solve problems involving arrangements or selections of items.

Combinations

Combinations are a fundamental aspect of combinatorics that deal with selecting items from a group, where the order does not matter. This is different from permutations, where order does matter. For example, when forming a committee of four people from a group of ten, we're interested in the number of different groups, not the arrangements within those groups.

The formula for calculating combinations is given by:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \),
• where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

We encounter combinations in various real-world situations, such as lottery draws or forming a panel of judges. Understanding combinatorics forms the basis for solving problems that involve possibilities, choices, and arrangements without repetition.

Algorithm Design

Algorithm design is the process of defining a step-by-step strategy to solve a specific problem, and it's critical in computer science for optimizing solutions. When designing algorithms for combinatorial problems such as calculating combinations, recursion is often employed because of its intuitive breakdown of larger problems into smaller ones.

Key considerations include:

• Efficiency: How quickly can the algorithm arrive at the solution? Recursive algorithms should minimize redundant calculations using memoization or iterative methods when applicable.
• Clarity: An algorithm should be easy to understand and maintain. Writing it clearly in pseudo-code first can aid in this.
• Correctness: Ensure that the algorithm accurately solves the problem. Test with known base cases and validate the results.

In our combination problem, a recursive algorithm is used to calculate \( C(n, r) \), by breaking the problem down into simpler sub-problems, making it an efficient and elegant solution.