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Chapter 4: Problem 13

Evaluate each expression. $$ \begin{array}{llll}{a .11 !} & {e ._{6} P_{4}} & {i ._{9} P_{2}} \\ {b .9 !} & {f ._ {12} P_{8}} & {j ._{11} P_{3}} \\ {c .}{0 !} & {g ._{7} P_{7}} \\ {d .} {1 !} & {h ., 70}\end{array} $$

Short Answer

Expert verified
a. 39916800, b. 362880, c. 1, d. 1, e. 360, f. 19958400, g. 5040, h. 70, i. 72, j. 990

Step by step solution

01

Calculate Factorials

Factorials are represented by '!'. They imply multiplying all integers from 1 up to the specified number. For example, \( n! = n \times (n-1) \times ... \times 1 \). Let's calculate the factorials requested: - \( 11! = 39916800 \) - \( 9! = 362880 \) - \( 0! = 1 \) (by definition) - \( 1! = 1 \)
02

Evaluate Permutations

Permutations \( _{n}P_{r} \) represent arranging \( r \) objects out of \( n \) distinct items, calculated using the formula: \[_{n}P_{r} = \frac{n!}{(n-r)!} \] Let's compute each permutation: - \(_6 P_4 = \frac{6!}{(6-4)!} = \frac{720}{2} = 360\) - \(_{12} P_8 = \frac{12!}{(12-8)!} = 19958400\) - \(_9 P_2 = \frac{9!}{(9-2)!} = \frac{362880}{5040} = 72\) - \(_7 P_7 = \frac{7!}{(7-7)!} = 5040\) - \(_{11} P_3 = \frac{11!}{(11-3)!} = 990\)
03

Answer the Expression

Now, list down the evaluated expressions as shown in the original problem: - \( a. 11! = 39916800 \) - \( b. 9! = 362880 \) - \( c. 0! = 1 \) - \( d. 1! = 1 \) - \( e. _6 P_4 = 360 \) - \( f. _{12} P_8 = 19958400 \) - \( g. _7 P_7 = 5040 \) - \( h. . 70 \) (as given) - \( i. _9 P_2 = 72 \) - \( j. _{11} P_3 = 990 \)

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Key Concepts

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

Factorials
Factorials are a fundamental concept in mathematics, particularly in the field of combinatorics and number theory. The factorial of a non-negative integer, denoted by the symbol '!', is the product of all positive integers less than or equal to that number. In mathematical terms, the factorial of a number \( n \) is represented as:
\[ n! = n \times (n-1) \times (n-2) \times ... \times 1 \]
  • \( 0! \) is defined to be 1 by convention. This definition helps in simplifying many mathematical expressions and formulae.
  • Factorials grow very quickly. For example, \( 5! = 120 \) and \( 10! = 3,628,800 \).
  • They're crucial in permutations and combinations, as they calculate the total ways of arranging objects.
Understanding the growth of factorials can help in comprehending their application in various mathematical contexts.
Mathematical Expressions
Mathematical expressions involve combinations of numbers, variables, operations, and other symbols to represent a particular value or relationship. In mathematics, expressions must be evaluated to solve problems or equations.
For instance, evaluate the expression \( _nP_r \) using its formula. It requires calculating factorials first and then using division to find the arrangement count. This illustrates how different mathematical operations can be combined to solve real-world problems like arranging a set of items.
  • Understand whether operations should be performed sequentially and which ones take precedence (PEMDAS rule: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
  • Expressions can range from simple (using primary arithmetic operations) to complex (involving logarithms, exponents, or trigonometry).
  • Clear understanding of symbols and operations ensures accurate interpretation and evaluation of expressions.
Mathematical expressions form the backbone of problem-solving in mathematics and beyond.
Combinatorics
Combinatorics is a branch of mathematics concerned with counting, arrangement, and combination of objects. It plays a vital role in numerous applications across mathematics, computer science, and statistics.
Permutations and combinations are primary tools in combinatorics:
  • Permutations: Concerned with the arrangement of objects where order matters. \(_nP_r\) is used to calculate the number of ways to arrange \( r \) objects out of \( n \).
  • Combinations: Involves selecting objects where order doesn't matter, calculated with a different formula \( _nC_r \).
  • These principles are used widely, from designing experiments to computer algorithms.
The ability to count efficiently and handle large numbers is invaluable in solving complex problems in both theoretical and practical domains.

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