Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
92 views
in Mathematics by (111k points)
closed by
How many different permutations can be made out of the letters of the word 'GIRLFRIEND'
1. \(\rm \frac{8!}{2!\;\times\; 2!}\)
2. \(\rm \frac{10!}{2!}\)
3. \(\rm \frac{10!}{2!\;\times \;2!}\)
4. \(\rm \frac{8!}{2!}\)

1 Answer

0 votes
by (105k points)
selected by
 
Best answer
Correct Answer - Option 3 : \(\rm \frac{10!}{2!\;\times \;2!}\)

Concept:

  • The ways of arranging n different things = n!
  • The ways of arranging n things, having r same things and rest all are different = \(\rm n!\over r!\)
  • The no. of ways of arranging the n arranged thing and m arranged things together = n! × m!
  • The number of ways for selecting r from a group of n (n > r) = nCr 
  • To arrange n things in an order of a number of objects taken r things = nPr  


Calculation:

The total number of words in GIRLFRIEND is 10

The word "I" in GIRLFRIEND repeated twice

Also, the word "R" GIRLFRIEND repeated twice

So, Number of different permutations = \(\rm \frac{10!}{2!\times 2!}\)

Permutation: Permutation is a way of changing or arranging the elements or objects in a linear order.

The number of permutations of 'n' objects taken 'r' at a time is determined by the following formula:

nP\(\rm \frac{n!}{(n - r)!}\)

nP= permutation

n = total number of objects
r = number of objects selected

The factorial function (Symbol: !just means to multiply a series of descending natural numbers.

For examples:

4! = 4 × 3 × 2 × 1 

1! = 1

There are three types of permutation:

  1. Permutations with Repetition
  2. Permutations without Repetition
  3. Permutation when the objects are not distinct (Permutation of multi-sets)


Representation of Permutation:

We can represent in many ways such as: 

  • P (n, k)
  • \(\rm P_{k}^{n}\)
  • n Pk
  • Pk 
  • P n, k

Application of Permutations:

  • Permutations are important in a variety of counting problems (particularly those in which order is important).
  • Permutations are used to define the determinant.

Order is very important in permutation.

"A Permutation is an ordered combination."

Permutation Combination
Permutation means the selection of objects, where the order of selection matters The combination means the selection of objects, in which the order of selection does not matter.
In other words, it is the arrangement of r objects taken out of n objects.  In other words, it is the selection of r objects taken out of n objects irrespective of the object arrangement.
The formula for permutation is nPr =   \(\rm \frac{n!}{(n - r)!}\)

 

The formula for combination is 

nCr =  \(\rm \frac{n!}{r!(n - r)!}\)

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...