Problem 48 - Algebra and Trigonometry

Question

These problems involve combinations. Choosing a Group In how many ways can six people be chosen from a group of ten?

Step-by-Step Solution

Answer

There are 210 ways to choose 6 people from a group of 10.

Step 1: Understanding the Problem

We need to select 6 people from a group of 10. This is a typical combination problem where order does not matter. Thus, we use the combinations formula.

Step 2: Choosing the Combinations Formula

Recall that the formula for combinations is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number, and \( r \) is the number of selections.

Step 3: Plug Values into the Formula

For this problem, \( n = 10 \) and \( r = 6 \). Substitute into the formula: \[ \binom{10}{6} = \frac{10!}{6!(10-6)!} \].

Step 4: Simplify Factorials

Calculate the factorials: \( 10! = 10 \times 9 \times 8 \times 7 \times 6! \) and \( 4! = 4 \times 3 \times 2 \times 1 \). This simplifies to:\[ \frac{10 \times 9 \times 8 \times 7 \times 6!}{6! \times 4 \times 3 \times 2 \times 1} \].

Step 5: Cancel Out Common Terms

Cancel \( 6! \) from both numerator and denominator, leaving:\[ \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} \].

Step 6: Calculate the Final Result

Multiply the remaining numbers in the numerator:\( 10 \times 9 = 90 \),\( 90 \times 8 = 720 \),\( 720 \times 7 = 5040 \).Then divide by the denominator:\[ \frac{5040}{24} = 210 \].

Key Concepts

Factorials

Factorials are a core component of calculating combinations and permutations. A factorial, denoted by an exclamation mark (!) after a number, means you multiply that number by every whole number below it. For instance, the factorial of 4 (written as 4!) is calculated as follows:

• 4! = 4 × 3 × 2 × 1 = 24.

Factorials are particularly useful in combinatorics because they account for all the different ways elements can be arranged. In many math problems related to combinations, you'll often see factorials in the formulas, such as in the 'n choose r' formula. Understanding how to simplify factorials by canceling common terms from numerator and denominator can greatly simplify complex calculations.

Combinatorics

Combinatorics is a branch of mathematics that deals with counting combinations, permutations, and other ways of arranging and selecting items from a set. It's particularly useful in problems where order does not matter, such as selecting a subset of items from a larger set. In the context of the exercise, combinatorics is applied to decide how many possible ways there are to pick a group of people (specifically 6 people) from a broader set (10 people).

• Understanding combinatorics helps solve problems involving selections without regard to order.
• It introduces important concepts like combinations and permutations which are fundamental in statistical calculations and probability.

Mastering combinatorics allows you to quickly determine the number of possible groupings, leading to an efficient problem-solving process in situations like the given exercise.

n choose r formula

The "n choose r" formula is a mathematical expression used to calculate the number of ways to choose r items from n total items, where the order of selection does not matter. The formula is:
\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]
In this formula:

• \( n \) = Total number of items from which selections are made.
• \( r \) = The number of items to choose.
• \binom{n}{r} = The number of combinations possible.

The formula accounts for eliminating the repetitive sequences that occur when the order of items is not significant. For example, if you apply this to the problem of choosing 6 people out of 10, you use:
\[\binom{10}{6} = \frac{10!}{6! \, 4!} = 210\]
This result reveals there are 210 different combinations possible, reaffirming that the order of selection doesn't matter in this scenario.

Permutations vs Combinations

Permutations and combinations are two foundational concepts which involve different ways of arranging or selecting items from a set.
Permutations are arrangements where the order of elements is important. For instance, the code for an alarm system is a permutation because the sequence of numbers matters. A simple formula for permutations of n items taken r at a time is:
\[P(n, r) = \frac{n!}{(n-r)!}\]
Combinations, on the other hand, focus on selection over arrangement, meaning the order does not matter. This is what you use when selecting committee members from a group or picking cards from a deck. It uses the "n choose r" formula discussed previously because it eliminates duplicates in which the same items appear in different sequences.

• Permutations = Arrangements (order matters).
• Combinations = Selections (order does not matter).

Knowing when to apply each concept is crucial for solving various mathematical and real-world problems.