Permutation Formula and Logarithm Formula

Permutation Formula

The permutation is the arrangement of objects in some definite order. Do you wonder how the schedules of trains and busses are made to suit our convenience? The concept of permutation and the permutation formula comes to the rescue. Also when we see the number plates of vehicles which consist of few alphabets and digits. We can easily prepare these codes using permutations. In this article, we will see the definition of permutation and permutation formula. Let us see the concept.

Permutation Formula

What is Permutation?

A permutation is a very important computation in mathematics. It is an arrangement of all or part of a set of objects, with regard to their order of the arrangement. Actually, very simply put, a permutation is an arrangement of objects in a particular way.

While dealing with permutation we should concern ourselves with the selection as well as the arrangement of the objects. Thus, ordering is very much essential in permutations.

For example, suppose we have a set of three letters: P, Q, and R. We have to find the number of ways we can arrange two letters from that set. Each possible arrangement will be one example of permutation. The complete list of possible permutations is PQ, PR, RP, QR, RP, and RQ.

When they refer to permutations, mathematicians use specific terminology. They describe permutations as an event when n distinct objects taken r at a time. Here, translation n refers to the number of objects from which the permutation is formed. Also, r refers to the number of objects used to form the permutation.

Consider the example given above. The permutation was formed from 3 alphabets (P, Q, and R),

So, n = 3;

Permutation consisted of 2 letters,

so r = 2.

Permutation Formula

The number of permutations of n objects, when r objects will be taken at a time.

nPr=(n) × (n-1) × (n-2) × …..(n-r+1)

i.e. nPr =n!(n−r)!

Here n! is the Factorial of n. It is defined as:

n!= (n) × (n-1) × (n-2) ×…..3 × 2 × 1

Other notation used for permutation: P(n,r)

In permutation, we have two main types as one in which repetition is allowed and the other one without any repetition. And for non-repeating permutations, we can use the above-mentioned formula.

For the repeating case, we simply multiply n with itself the number of times it is repeating. It means that nr, where n is the number of things to be chosen from and r, is the number of items being chosen.

Solved Examples

Q. How many 3 letter words with or without meaning can be framed out of the letters of the word SWING? Repetition of letters is not allowed?

Solution: Here n = 5, because the number of letters is 5 in word SWING.

Since we have to frame words of 3 letters without repetition.

Therefore permutations possible are:

P(n,r)

= 5!(5−3)!

= 60

Q. How many 3 letter words with or without meaning can be created out of the letters of the word SMOKE. Note that the repetition of letters is allowed?

Solution:

The number of objects, here is 5, because the word SMOKE has 5 alphabets.

Also, r = 3, as 3 letter-word has to be chosen.

Thus the permutation will be

= nr as repetition is allowed.

= 53

= 125

Logarithm Formula

Logarithms are widely used in computations in mathematics as well as in science. It helps to solve complex problems involving exponents of variables, easily. Many derivations of physics are possible only due to Logarithm Formula. A logarithm is the inverse computation process of exponential. Logarithms are widely used in the field of physics, chemistry, biology, computer, etc. We can even find logarithmic calculators which have made our calculations much faster and easier. These find many applications in surveying and celestial navigation purposes. To understand the concept of the logarithm, let us have this useful article. We will see various logarithm formula in mathematics with examples and their applications.

Logarithm Formula

What is Logarithm?

As we know that 34=81

Now suppose if we are asked the same question but differently, like “what will be the exponent of 3 to get the result 81?”

Then obviously answer will be 4. But how? The answer to this question only is the basic definition of logarithms.

Now, we will write the above equation in the form of a logarithm, as

log3 81 = ?

Here, 3 is the base whose exponent we have to find. So we wish to find the value which when rose as the power to 3 will be equal to 81. Since this will be 4, so we will say that

log381 = 4

This above equation will be read as “log base 3 of 81 is 4”.

Thus ,general definition and rule of logarithm is:

logax=b⇒x=ab.

Hence, the exponent or power to which a base must be raised to yield a given number is nothing but the logarithm.

Also,

Since 10² = 100,

Then we get

2=log10100

Logarithms with the base 10 are called common or Briggsian logarithms and it is written simply log n. It is invented in the 17th century to speed up calculations. The natural Logarithm is with base e where e ≅ 2.71828, and it is written as ln n.

Logarithm Formula:

Two most trivial identities of logarithms are:

(1) logb1=0

This is because  b0 = 1;

(2) if b>0 then

logbb=1

This is because b1=b

Some other very important formula are:

Suppose a, b , m, n are variables with positive integers and p as a real number. Then we have,

(1) logbm∗n=logbm+logbn

(2) logbmn=logbm+logbn

(3) logbnp=plogbn

(4) logbn=logan∗logba

Solved Examples

Q.1: Solve log64 = ?

Solution- since 26=2∗2∗2∗2∗2∗2

i.e. 26 = 64,

Thus 6 is the exponent value.

So, log264 = 6.

Q.2:  By using property of logarithms, solve for the value of x

log3x=log34+log37

Solution- Here for RHS we will use the addition rule of logarithms.

log3x=log34+log37

Then \(\log_3 x = \ log_3(4 * 7 )\

Thus value of x is = 28.