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#1 2020-01-07 01:36:30

666 bro
Member
From: Flatland
Registered: 2019-04-26
Posts: 706

Factorials

How to find the value of (n-a)! Where "a" is any number?


"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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#2 2020-01-07 01:58:15

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,955

Re: Factorials

Hi 666 bro,

Hope this helps:

Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.

There are n! different ways of arranging n distinct objects into a sequence, the permutations of those objects.

Often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations (subsets of k elements) from a set with n elements. One can obtain such a combination by choosing a k-permutation: successively selecting and removing one element of the set, k times, for a total of

possibilities. This, however, produces the k-combinations in a particular order that one wishes to ignore; since each k-combination is obtained in k! different ways, the correct number of k-combinations is

This number is known as the binomial coefficient, because it is also the coefficient of

in
. The term
is often called a falling factorial (pronounced "n to the falling k").

Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.

Factorials also turn up in calculus; for example, they occur in the denominators of the terms of Taylor's formula, where they are used as compensation terms due to the nth derivative of

being equivalent to n!.

Factorials are also used extensively in probability theory and number theory.

Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n can be written as

while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:

Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.

Factorials also turn up in calculus; for example, they occur in the denominators of the terms of Taylor's formula,[9] where they are used as compensation terms due to the nth derivative of

being equivalent to n!.

Factorials are also used extensively in probability theory and number theory.

Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n can be written as


while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients:


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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