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Questions
\(\textbf{1)}\) \( 5! \)
\(\textbf{2)}\) \( 4! \)![Link to Youtube Video Solving Question Number 2](https://andymath.com/wp-content/uploads/2020/08/play-video.jpg")
\(\textbf{3)}\) \( 1! \)
\(\textbf{4)}\) \( 0! \)
\(\textbf{5)}\) \( \frac{16!}{12!4!} \)
\(\textbf{6)}\) \( \frac{19!}{18!} \)
\(\textbf{7)}\) \( \frac{8!}{6!} \)
Challenge Problems
\(\textbf{8)}\) \( \frac{n!}{(n-2)!} \)
\(\textbf{9)}\) \( \frac{n!}{n} \)
\(\textbf{10)}\) \( \frac{x}{x!}=50\% \)
\(\textbf{11)}\) \( 6!+7!=x! \)
See Related Pages\(\)
\(\bullet\text{ Statistics Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Combinations and Permutations}\)
\(\,\,\,\,\,\,\,\,{}_n{C}_r=\displaystyle\frac{n!}{r!(n-r)!},\,\,\,{}_n{P}_r=\displaystyle\frac{n!}{(n-r)!}…\)
In Summary
The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. It is denoted by the symbol n!. For example, the factorial of 4, denoted as 4!, is equal to 4 x 3 x 2 x 1, which is equal to 24.
We learn about factorials in mathematics because they have many practical applications in fields such as statistics and combinatorics. Factorials are used to calculate the number of ways that a given number of objects can be arranged, and they are an important part in finding the probability of certain events occurring.
Factorials are typically introduced in high school math classes, often in the context of algebra or pre-calculus. However, they can also be encountered in lower-level math classes, depending on the curriculum.
Did you know that the factorial of 100 is equal to a number with 158 digits? This number is so large that it’s almost impossible to comprehend. In fact, it’s much larger than the estimated number of atoms in the observable universe!
The concept of factorials was first introduced by a French mathematician named Blaise Pascal in the 17th century. However, the notation n! was not used until the 19th century, when it was introduced by the mathematician Christian Kramp.
Math topics that use factorials
Biology: Factorials are used to calculate the number of different possible arrangements of a set of genetic traits.
Chemistry: Factorials are used to calculate the number of ways in which a set of molecules can be arranged. For example, if a molecule has 4 different groups of atoms attached to it, there are 4! = 24 possible arrangements of those groups.
Permutations: Factorials are often used in counting the number of permutations of a set of objects. For example, if you have a set of 3 objects (let’s say, A, B, and C), there are 3! = 6 possible permutations of those objects: ABC, ACB, BAC, BCA, CAB, CBA.
Combinations: Factorials are also used in counting the number of combinations of a set of objects. Combinations are a way of selecting objects from a set without regard to order, so ABC is considered the same as ACB, BAC, BCA, CAB, and CBA. The number of combinations of a set of n objects is given by the formula: \(\frac{n!}{(r!(n-r)!)}\), where r is the number of objects selected from the set.
Probability: Factorials are used in probability calculations, particularly when dealing with combinations of events.
Discrete mathematics: Factorials arise in a number of areas of discrete mathematics, such as graph theory and combinatorics. In graph theory, factorials are used in counting the number of ways to arrange the vertices of a graph in a particular way, such as in a Hamiltonian cycle (a cycle that passes through every vertex exactly once). In combinatorics, factorials are used in counting the number of ways to arrange objects in a given structure, such as a Young tableau (a way of arranging objects in a grid).
Calculus: Factorials also appear in calculus, particularly in the development of Taylor series and power series. Taylor and power series are an infinite series expansion of a function, and the coefficients of the series can be expressed in terms of factorials.
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