# Combination & Logarithm

**Combination**

The selection of articles from a group of articles without repetition in which the order does not matter can be termed as combination whereas permutations refer to the arrangement of objects in a definite order. It can be represented by ^{n}C_{k} = n! / (n – k)! k !. The field of mathematics that deals with problems involving counting are Combinatorics. The basic principle of counting includes the fundamental principle and the addition principle.

Some of the important formulae in Combination are as follows.

^{n}C_{n }= n! / [n! 0!] = 1^{n}C_{k}= n! / (n – k)! k !, 0 ≤ k ≤ n^{n}C_{n-r}= n! / (n – r)! [n – (n – r)]! =^{n}C_{r}^{n}C_{a}=^{n}C_{b}^{n}C_{r}+^{n}C_{r-1}=^{n+1}C_{r}

For example, how many cards each of 5 can be possible from a standard fifty-two card deck?

**Solution: **

^{52}C_{5} = [52 * 51 * 50 * 49 * 48] / [5 * 4 * 3 * 2 * 1] = 311875200 / 120 = 2598960

**Relation between permutations and combinations**

The concept of combination is one of the kinds of permutations where the order of the selection is not taken into consideration. The total of permutation is always > the number of the combinations. This is the basic difference between them.

**Theorem: **^{n}P_{r} = ^{n}C_{r }. r!

^{n}P

_{r}=

^{n}C

_{r }. r!

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In every combination of ^{n}C_{r}, r! permutations can be possible since r objects can be rearranged in r! ways in every combination.

**Proof:**

^{n}P_{r} = ^{n}C_{r}.r!

= [n! / r! (n – r)!] . r!

= n! / (n – r)!

The theorem holds good.

**Logarithm**

The function which is the inverse of exponentiation can be termed as a logarithm. It can be explained as “the logarithm of any given number a is the exponent of another number that is fixed, with base b, must be an exponent to obtain the number a.” The topic of logarithms was introduced by John Napier in 1614 as a method to simplify calculations. The logarithm of x with base b is denoted as log_{b} x. Here, b and x are real numbers, b ≠ 1. The logarithm with base 10 is the decimal or common logarithm. The concept of the binary logarithm that is with base 2 can be found in computer science. The different parts of a logarithm include the mantissa and characteristic. Characteristics is an integral part of a logarithm whereas mantissa is the decimal part. It is denoted as follows.

**log N = Integer + Fractional or Decimal Part (Positive)**

Properties of logarithms are given below.

- log
_{a}(mn) = log_{a}m + log_{a}n - log
_{a}(m / n) = log_{a}m – log_{a}n - log
_{a}m^{q}= q log_{a}m - log
_{b}m = log_{a}m / log_{a}b - log
_{b}a . log_{a}b = 1 ⇒ log_{b}a = 1 / log_{a}b - log
_{b }a . log_{c }b . log_{a}c = 1 - log
_{y }x . log_{z }y . log_{a}z = log_{a }x - e
^{ln}^{a^x}= a^{x}

Applications of logarithms are listed below.

- The concept of logarithmic scales is used in reducing the wide-range quantities of tiny scopes.
- It is used in measuring the acidity of an aqueous solution called pH.
- It can be used to measure the complexity of algorithms and geometric objects called fractals.
- They help in explaining the frequency ratio of intervals of music.
- It also aids in forensic accounting.
- The modular discrete logarithm is used in public-key cryptography.

Some of the applications of algorithms are listed above for reference. For more information on the above topic, please refer to BYJU’S.