How many different ways are there to arrange letters of the word WORLD ? How many of these arrangements begin with letter R? How many arrangements can be made taking three letters at a time ?
Answers
Concept:
We recall the concept of permutation before proceeding to the solution for this question.
A permutation is the different arrangements that can be made out of a given number of things by taking some or all of them at a time.
This is a word problem on permutations.
Given:
We are given the word 'WORLD'
To find:
We are asked to find:
- No. of different ways we can arrange the word given
- No. of arrangements which begin with the letter R
- No. of arrangements with three letters at a time
Solution:
The given word WORLD has 5 letters.
Therefore,
The required no. of arrangements of the word
=Number of arrangements of 5 letters taking 5 of them at a time
=5!
=5x4x3x2x1
=120
Now, for the second part, if we fix the letter R in the beginning.
Then, the letters will be considered 4 which are to be arranged as one position remains fixed.
∴The required no. of arrangements will be
= 4!
=4x3x2x1
=24
Now, if we take three letters together we will count them as one during the arrangement, so the no of letters to be arranged will count down to 3 and the three letters which are taken together also will be arranged among themselves
Hence,
Required no. of arrangements taking three letters together
=3!x3!
=(3x2x1)x3!
=6x(3x2x1)
=6x6
=36
Conclusion:
- No. of arrangements of the word=120
- No. of arrangements taking R at beginning=24
- No. of arrangements taking 3 letters together=36
Answer:
Concept:
a permutation of a set is a loose arrangement of its members into a sequence or linear order, or a rearrangement of its elements if the set is already sorted. The act or process of changing the linear order of an ordered set is also known as permutation.
Given:
Given word World
Find:
No. of ways to arrange the word world
No. of ways to arrange the word with begins with letter R
No. of ways to arrange the word with three letters at a time
Step-by-step explanation:
Given word = world
1: Number of arrangements of the word world
Number of letters = 5
= 5!
= 5×4×3×2×1
= 120
∴ The Number of arrangements of the word world = 120
2: Number of arrangements when the word begins with R
If the letter is fixed then we can consider 4 letters
letters to be arranged = 4!
= 4×3×2×1
= 24
∴ The Number of arrangements when the word begins with R = 24
3: Number of arrangements taking 3 letters at a time
number of arrangements = 3!×3!
= (3×2×1)×(3×2×1)
= 6×6
= 36
∴ Number of arrangements taking 3 letters at a time = 36