Question

Evaluate the expression. \({ }_8 C_6\)

Short answer

The value of \({ }_8 C_6\) is 28.

Step-by-step solution

Substituting the values into the formula

In our problem, we have \(n = 8\) and \(r = 6\). So, \({ }_8 C_6 = \frac{8!}{6!(8-6)!}\). This simplifies to \({ }_8 C_6 = \frac{8!}{6!2!}\)

Simplifying the factorials

We can simplify the factorials. The \(8!\) equals to \(8 * 7 * 6 * 5 * 4* 3* 2 * 1\). The \(6!\) equals to \(6 * 5 * 4 * 3 * 2 * 1\). And the \(2!\) is equal to \(2 * 1\). So, \({ }_8 C_6 = \frac{8 * 7 * 6!}{6! * 2* 1}\)

Cancelling out similar terms

The \(6!\) in the numerator and the denominator cancels away. So, we are left with \({ }_8 C_6 = 28\).

Final Answer

After simplifying the combination, we find that \({ }_8 C_6 = 28\).

Key concepts

Factorials

Factorials are a foundational concept in combinatorics. The factorial of a non-negative integer \(n\) is denoted by \(n!\) and represents the product of all positive integers from 1 to \(n\).
For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). Factorials grow very quickly as \(n\) increases, which is why simplifying and canceling them in equations is a common practice.
In combinatorial problems, factorials are often used to calculate the total number of ways to arrange or choose items. They form the basis for understanding other concepts such as permutations and combinations.
Factorials can be simplified in expressions to make calculations manageable. For instance, dividing \(8!\) by \(6!\) in the combination formula helps eliminate repetitive terms, simplifying the computation.

Combinations

Combinations are ways of selecting items from a larger pool, where the order does not matter. When we calculate combinations, we use the formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose.
This formula helps find how many possible selections there are. For example, \({ }_8 C_6\) means selecting 6 items from a total of 8, disregarding the order.
Combinations focus on selection rather than arrangement, distinguishing them from permutations.

• Key features of combinations include the ability to choose subsets of a set without repetition.
• In practical scenarios, you might use combinations to figure out lottery odds or to determine the number of ways to form groups from a class.

The main idea is how many different ways items can be selected.

Permutations

Permutations refer to the arrangements of items where the order is crucial. The formula for calculating permutations of \(r\) items from \(n\) is \(P(n, r) = \frac{n!}{(n-r)!}\).
This calculation gives us the number of different ways \(r\) items can be ordered from a total of \(n\).
Unlike combinations, which deal with selections, permutations consider each specific ordering distinct.
For instance, in permutations, arranging the letters A, B, and C results in different outcomes: ABC, BAC, CAB, etc.

• Permutations are essential in situations where arrangement sequence matters, such as in password generation or scheduling.
• Distinguishing between permutations and combinations is vital to solve combinatorial problems accurately.

They help identify how many distinct ways items can be ordered.

Binomial Coefficient

The binomial coefficient, usually expressed as \(\binom{n}{r}\), is a mathematical term used to represent the number of ways \(r\) elements can be chosen from \(n\) elements, regardless of the order.
It is a fundamental component of the binomial theorem and appears in the expansion of binomial expressions.
The formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) is used, matching the combinations formula, highlighting that binomial coefficients are the building blocks of combinations.

• In algebra, binomial coefficients are used in polynomial expansions calculated using Pascal's triangle or directly through the formula.
• They are essential in probability and statistics, often seen in binomial distribution calculations.

Understanding binomial coefficients is key to mastering both theoretical and applied aspects of combinatorics.