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Chapter 9: Problem 82

Solve for \(n\) $$_{n} P_{6}=12 \cdot_{n-1} P_{5}$$

Short Answer

Expert verified
We can apply the factorial rule only when \(n\) is a non-negative integer. So with that constraint the solution to the equation is \(n\) equals 7 or 12.

Step by step solution

01

Apply Permutations Formula

Permutations formula is given as - \( P(n,k) = \frac{n!}{(n-k)!} \) Let's apply this formula to our problem. For the left side, we substitute the formula, it is going to be \(\frac{n!}{(n-6)!}\).Applying the same rule to the right-hand side the formula converts to \(12 \cdot \frac{(n-1)!}{(n-6)!}\).
02

Equation simplification

Now simplify the equation by cancelling out the common terms i.e., \((n-6)!\) from both sides and the equation is simplified to \(n! = 12 \cdot (n-1)!\)
03

Find Possible values of \(n\)

To find the values of \(n\), one can start to substitute values starting from 6 because anything less than 6 makes the factorial negative which isn't defined. After trying with a few numbers, you'll find that the equation balances out when \(n\) equals to 7 or 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorial
Factorials are a fundamental concept in permutations and combinations. They are used to count the number of ways to arrange a set of items. The factorial of a non-negative integer, denoted by the symbol \( n! \), 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 grow very quickly with increasing \( n \). This makes them extremely useful for calculations involving large sets of elements.

Key points about factorials:
  • \( 0! \) is defined as 1.
  • Factorials are used extensively in probability, statistics, and algebra.
  • They provide a way to easily count large quantities, like the number of permutations or combinations of a set.
Equation Simplification
Equation simplification involves rewriting equations in a simpler form. This is essential in mathematics to make equations easier to work with or solve. In the context of permutations and factorials, simplification often involves cancelling out common terms on both sides of the equation.

For example, in the given problem, we have the equation \( \frac{n!}{(n-6)!} = 12 \times \frac{(n-1)!}{(n-6)!} \). The term \((n-6)!\) appears on both sides, and can thus be cancelled, simplifying the equation to \( n! = 12 \times (n-1)! \).

Steps to Simplify an Equation:
  • Identify and cancel common factors on both sides.
  • Combine like terms if possible.
  • Use basic arithmetic operations to simplify further.
  • Ensure each step maintains the equivalence of the equation.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. It helps answer questions about the number of possible arrangements or selections of a set of items. Permutations are a common topic within combinatorics which deal with the arrangement of objects in a specific order.

In permutations, we calculate how many different ways we can order a subset of items from a larger set. The formula for permutations, \( P(n,k) = \frac{n!}{(n-k)!} \), allows us to compute this number based on the factorial of the total and remaining items in the set.

Important Aspects of Combinatorics:
  • Permutations: Focused on order, used when arrangement matters.
  • Combinations: Focused on selection, used when arrangement doesn't matter.
  • Used in fields like computer science, logistics, and probability.
  • Helps solve real-world problems involving arrangement and selection.

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Most popular questions from this chapter

Find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Pentagon

Prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n\).$$_{n} C_{0}-_{n} C_{1}+_{n} C_{2}-\cdots \pm_{n} C_{n}=0$$.

Use the Binomial Theorem to expand the complex number. Simplify your result. $$(5+\sqrt{-9})^{3}$$

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form.$$f(x)=-x^{4}+4 x^{2}-1, \quad g(x)=f(x-3)$$.

Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise \(79,\) use the expansion \(\begin{aligned}(1.02)^{8} &=(1+0.02)^{8} \\ &=1+8(0.02)+28(0.02)^{2}+\cdot \cdot \cdot+(0.02)^{8}\end{aligned}\), $$(2.99)^{12}$$
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