Find permutations of letters taken all at a time that can be formed out of 'watch
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The different ways in which the alphabets A, B and C can be grouped together, taken all at a time, are ABC, ACB, BCA, CBA, CAB, BAC.
Note that ABC and CBA are not same as the order of arrangement is different. The same rule applies while solving any problem in Permutations.
The number of ways in which n things can be arranged, taken all at a time, nPn = n!, called ‘n factorial.’
Factorial Formula
Factorial of a number n is defined as the product of all the numbers from n to 1.
For example, the factorial of 5, 5! = 5*4*3*2*1 = 120.
Therefore, the number of ways in which the 3 letters can be arranged, taken all a time, is 3! = 3*2*1 = 6 ways.
Number of permutations of n things, taken r at a time, denoted by:
nPr = n! / (n-r)!
For example:
The different ways in which the 3 letters, taken 2 at a time, can be arranged is 3!/(3-2)! = 3!/1! = 6 ways.
Important Permutation Formulas
1! = 1
0! = 1
Let us take a look at some examples:
Problem 1: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘CHAIR’.
Solution:
‘CHAIR’ contains 5 letters.
Therefore, the number of words that can be formed with these 5 letters = 5! = 5*4*3*2*1 = 120.
Problem 2: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘INDIA’.
Solution:
The word ‘INDIA’ contains 5 letters and ‘I’ comes twice.
When a letter occurs more than once in a word, we divide the factorial of the number of all letters in the word by the number of occurrences of each letter.
Therefore, the number of words formed by ‘INDIA’ = 5!/2! = 60.
Problem 3: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘SWIMMING?
Solution:
The word ‘SWIMMING contains 8 letters. Of which, I occurs twice and M occurs twice.
Therefore, the number of words formed by this word = 8! / (2!*2!) = 10080.
Note that ABC and CBA are not same as the order of arrangement is different. The same rule applies while solving any problem in Permutations.
The number of ways in which n things can be arranged, taken all at a time, nPn = n!, called ‘n factorial.’
Factorial Formula
Factorial of a number n is defined as the product of all the numbers from n to 1.
For example, the factorial of 5, 5! = 5*4*3*2*1 = 120.
Therefore, the number of ways in which the 3 letters can be arranged, taken all a time, is 3! = 3*2*1 = 6 ways.
Number of permutations of n things, taken r at a time, denoted by:
nPr = n! / (n-r)!
For example:
The different ways in which the 3 letters, taken 2 at a time, can be arranged is 3!/(3-2)! = 3!/1! = 6 ways.
Important Permutation Formulas
1! = 1
0! = 1
Let us take a look at some examples:
Problem 1: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘CHAIR’.
Solution:
‘CHAIR’ contains 5 letters.
Therefore, the number of words that can be formed with these 5 letters = 5! = 5*4*3*2*1 = 120.
Problem 2: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘INDIA’.
Solution:
The word ‘INDIA’ contains 5 letters and ‘I’ comes twice.
When a letter occurs more than once in a word, we divide the factorial of the number of all letters in the word by the number of occurrences of each letter.
Therefore, the number of words formed by ‘INDIA’ = 5!/2! = 60.
Problem 3: Find the number of words, with or without meaning, that can be formed with the letters of the word ‘SWIMMING?
Solution:
The word ‘SWIMMING contains 8 letters. Of which, I occurs twice and M occurs twice.
Therefore, the number of words formed by this word = 8! / (2!*2!) = 10080.
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