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Magazine Chapter 5: Problem 41
Calculate the given combination.$$_{10}C_{6}$$
Short Answer
Expert verified
210
Step by step solution
01
Understand the Combination Formula
The number of combinations of selecting a group of items from a larger set can be calculated using the formula: \[ _nC_k = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, \( k \) is the number of items to choose, and \(!\) denotes factorial.
02
Identify the Values of n and k
For this problem, \( n = 10 \) and \( k = 6 \). We need to calculate \( _{10}C_6 \).
03
Calculate Factorials
First, calculate the factorial of each necessary number. We have:* \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800 \)* \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)* \( 4! = 4 \times 3 \times 2 \times 1 = 24 \), since \( 4 = 10 - 6 \).
04
Substitute Values into Combination Formula
Substitute the values of the factorials into the formula:\[ _{10}C_6 = \frac{10!}{6!(10-6)!} = \frac{3628800}{720 \times 24} \]
05
Simplify the Expression
Calculate the denominator \( 720 \times 24 = 17280 \). Now, simplify the whole expression:\[ \frac{3628800}{17280} = 210 \]
06
State the Final Result
The number of combinations for choosing 6 items from 10 is 210.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factorials
A factorial is a mathematical operation that multiplies a series of descending natural numbers. When we talk about the factorial of a number, denoted as \( n! \), it represents 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 are essential when dealing with permutations and combinations as they form the basis of calculating possibilities in various scenarios.
Here鈥檚 why we use factorials in combinations:
Factorials are essential when dealing with permutations and combinations as they form the basis of calculating possibilities in various scenarios.
Here鈥檚 why we use factorials in combinations:
- They help in calculating the different ways to arrange or select items.
- Factorials give us the total number of ways to order a set number of items.
- In combinations, we're interested in the number of ways to choose items, which requires dividing one factorial by others.
Combination Formula
The combination formula helps us determine how many ways we can choose a subset from a larger set, without considering the order of selection. Represented as \( _nC_k \) or simply \( C(n, k) \), this formula is useful in various counting problems.
The formula itself, \( _nC_k = \frac{n!}{k!(n-k)!} \), breaks down as follows:
The formula itself, \( _nC_k = \frac{n!}{k!(n-k)!} \), breaks down as follows:
- Numerator (\( n! \)): This represents the total arrangement of \( n \) items.
- Denominator (\( k!(n-k)! \)): It eliminates the order, dividing by the arrangements of the chosen \( k \) items and the arrangement of items not chosen \((n-k)\).
Binomial Coefficients
Binomial coefficients are the numerical factors you encounter in the expansion of a binomial raised to a given power. Found in what is known as the binomial theorem, each coefficient represents the number of ways to choose a subset of elements from a set.
In expressions, binomial coefficients are represented as \( \binom{n}{k} \), resembling the notation\( _nC_k \), both indicating combinations without regard to order.
Their importance in combinations is highlighted due to:
In expressions, binomial coefficients are represented as \( \binom{n}{k} \), resembling the notation\( _nC_k \), both indicating combinations without regard to order.
Their importance in combinations is highlighted due to:
- Their ability to represent combinations effortlessly within binomial expansions.
- Providing a way to directly read off combo options from the binomial theorem's expansion.
- Helping in various probability problems, especially when distances, areas, and volumes are involved.
