Combinations - RD Sharma Solved Problems

 

What is a Combination?

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter. In combinations, you can select the items in any order.

Combinations can be confused with permutations. However, in permutations, the order of the selected items is essential. For example, the arrangements ab and ba are equal in combinations (considered as one arrangement), while in permutations, the arrangements are different.

Formula for Combination

Mathematically, the formula for determining the number of possible arrangements by selecting only a few objects from a set with no repetition is expressed in the following way:

 

 

Where:

• n – the total number of elements in a set
• k – the number of selected objects (the order of the objects is not important)
• ! – factorial

A few important results on combinations are as follows:

• The number of ways of selecting n objects out of n objects is:${}^n{C}_n=\frac{n!}{n!\left(n-n\right)!}=\frac{n!}{n!0!}=1$
• The number of ways of selecting 0 objects out of n objects is:${}^n{C}_0=\frac{n!}{0!\left(n-0\right)!}=\frac{n!}{0!n!}=1$
• The number of ways of selecting 1 object out of n objects is: ${}^n{C}_1=\frac{n!}{1!\left(n-1\right)!}=\frac{n\times \left(n-1\right)!}{\left(n-1\right)!}=n$
• ${}^n{C}_r{=}^n{C}_{n-r}$
• ${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$

Indian Statistical Institute : Solved Problems

TOMATO (TEST OF MATHEMATICS AT THE 10+2 LEVELS) 

OBJECTIVE PROBLEM SOLVING

 

ISI's Test of Mathematics at 10 + 2 Level is a rigorous book for mathematics enthusiasts and for Class-12 students applying for an admission into the Bachelor of Mathematics at one of the Indian Statistical Institutes across the nation. The book provides advanced problems that are commonly found in the ISI entrance examinations and also prepares students for a study involving higher mathematics.

About the Indian Statistical Institute

ISI is an academic institute in India catering to studies in statistics. It was established in 1931 to improve the application of statistical methods across the nation. The institute provides academic programs at the undergraduate and postgraduate level in mathematics, statistics and numerical methods.

PART -1 QUESTION 1 TO QUESTION 28

Permutations - RD Sharma Solved Problems.

 A permutation is an arrangement of objects in a definite order. The members or elements of sets are arranged here in a sequence or linear order. For example, the permutation of set A={1,6} is 2, such as {1,6}, {6,1}. As you can see, there are no other ways to arrange the elements of set A.

In permutation, the elements should be arranged in a particular order whereas in combination the order of elements does not matter. Also, read: Permutation And Combination

When we look at the schedules of trains, buses and the flights we really wonder how they are scheduled according to the public’s convenience. Of course, the permutation is very much helpful to prepare the schedules on departure and arrival of these. Also, when we come across licence plates of vehicles which consists of few alphabets and digits. We can easily prepare these codes using permutations.

A permutation is defined as an arrangement in a definite order of a number of objects taken, some or all at a time. Counting permutations are merely counting the number of ways in which some or all objects at a time are rearranged. The convenient expression to denote permutation is defined as “ nPr ”.

The permutation formula is given by,

Pr = n!/(n-r)! ; 0 ≤ r ≤ n

Where the symbol “!” denotes the factorial which means that the product of all the integers is less than or equal to n but it should be greater than or equal to 1.

Permutation When all the Objects are Distinct

There are some theorems involved in finding the permutations when all the objects are distinct. They are :

Theorem 1: If the number of permutations of n different objects taken r at a time, it will satisfy the condition 0 < r ≤ n and the objects which do not repeat is n ( n – 1) ( n – 2)……( n – r + 1), then the notation to denote the permutation is given by “ n Pr”

Theorem 2: The number of permutations of different objects “n” taken r at a time, where repetition is allowed and is given by nr .

Permutation When all the Objects are not Distinct Objects

Theorem 3: To find the number of permutations of the objects ‘n’, and ‘p’s are of the objects of the same kind and rest is all different is given as n! / p!

Theorem 4: The number of permutations of n objects, where p1 are the objects of one kind, p2 are of the second kind, …, pk is of the kth kind and the rest, if any, are of a different kind, then the permutation is given by n! / ( p1!p2!…Pk!)