Problem 11 - Precalculus with Limits

Question

In Exercises 5 - 14, calculate the binomial coefficient. \( \dbinom{10}{4} \)

Step-by-Step Solution

Answer

The binomial coefficient \( \dbinom{10}{4} \) is 210.

Step 1: Understanding Binomial Coefficient Formula

The binomial coefficient is calculated using the following formula: \( \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!} \), where '!' denotes factorial, multiplying all positive integers up to the number. Here n=10 and r=4.

Step 2: Substitute the values into the formula

Substitute n=10 and r=4 into the binomial coefficient formula. It becomes \( \dbinom{10}{4} = \dfrac{10!}{4!(10-4)!} \)

Step 3: Simplifying the equation

Now, simplify the equation. Calculate the factorials and then simplify the fraction: \( \dbinom{10}{4} = \dfrac{10*9*8*7*6*5*4*3*2*1}{4*3*2*1*6*5*4*3*2*1} \)

Step 4: Calculate the binomial coefficient

After simplification, divide the numerator by the denominator to find the binomial coefficient: \( \dbinom{10}{4} = 210 \)

Key Concepts

Factorial

A factorial is a fundamental concept in mathematics and is denoted by the symbol '!'. When you see a number followed by '!', it means you multiply that number by every positive integer below it. For example, 5 factorial, written as 5!, is calculated as 5 × 4 × 3 × 2 × 1, which equals 120.
Your main takeaway from this is that factorials involve multiplying a series of descending natural numbers. The factorial function grows very quickly, which makes it particularly useful in combinatorics where large number calculations are common.
Factorials are used to calculate permutations, combinations, and also appear in various areas of mathematics like calculus and algebra. For instance, 0! is defined to be 1 because it serves as a multiplicative identity, ensuring the consistency of formulas like the binomial coefficient.

Combinatorics

Combinatorics is the field of mathematics that studies counting and arrangement possibilities. It's a particularly useful branch when dealing with problems related to probability, statistics, and optimization.
Understanding combinatorics involves learning about different ways of arranging objects, selecting items from groups, or distributing objects into bins. Some fundamental concepts in combinatorics include permutations and combinations, each of which serves a different purpose in counting.
Combinatorics can be applied in real-world scenarios such as analyzing possible outcomes of experiments, planning the design of computer networks, and solving scheduling problems. Its applications are numerous and diverse, making it an important field for anyone who deals with situations requiring counting and arrangement strategies.

Permutations and Combinations

Permutations and combinations are two key concepts in combinatorics that help in finding the number of possible arrangements or selections of objects.

• Permutations: When order matters, permutations are used. For example, if you have 4 letters and you want to know in how many ways you can arrange them, you would calculate a permutation. The formula for permutations when selecting 'r' objects from a set of 'n' is \( P(n, r) = \dfrac{n!}{(n-r)!} \).
• Combinations: When order does not matter, combinations are used. This is useful for situations like selecting a team from a roster where the order in which you choose team members is irrelevant. The formula here is \( C(n, r) = \dfrac{n!}{r!(n-r)!} \), which is indeed what the binomial coefficient calculates.

Combinations are used in the solution to the exercise because the binomial coefficient is essentially a combination, denoting how many ways we can choose 4 items from a set of 10 without regard to order, resulting in 210 ways.