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Factorial

The factorial represents the product of multiplying all the positive integers from 1 up to a given integer. This concept is symbolized by an exclamation point after the given integer, such as $5!\$.

$5!\$   means   $5\cdot 4\cdot 3\cdot 2\cdot 1$

The factorial is used in a variety of fields of mathematics including probability and discrete counting.

Uses of factorial

The product of all the positive integers in a range other than 1 to n can be denoted and calculated by the division of two factorials. The formula for the product of the integers from k up to n is

${\frac {n!}{(n-k)!}}$

Example:

$13\cdot 12\cdot 11\cdot 10\cdot 9={\frac {13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}}$

The fraction in this formula will always reduce to an integer, because each of the factors in the denominator is also a factor of the numerator. The product of the factors remaining after the cancellation yields the desired result.

Binomial coefficient

In expanding the product: $(x+y)^{n}\$, the coefficient of each term is given by the formula

${\frac {n!}{k!\cdot (n-k)!}}$

where n is the exponent of the binomial in its original form, k and n-k are the exponents of the x and y in the expanded form

Example: In the expansion of $(x+y)^{7}\$ , the variable part of the third term is $x^{5}y^{2}\$ . The coefficient of this term is:

${\frac {7!}{5!\cdot 2!}}={\frac {7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 2\cdot 1}}={\frac {7\cdot 6}{2\cdot 1}}=21$

The fraction in this formula will always cancels completely into an integer. The larger of the two denominator factorials cancels as in the paragraph above. The other factors all cancel out because the same number of factors remain in the numerator as in the denominator. And, in any sequence of n consecutive integers