How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if (i) 4 letters are used at a time, (ii) all letters are used at a time, (iii) all letters are used but first letter is a vowel?
The word MONDAY contains 6 different letters.
(i)
The number of 4-letter words, with or without meaning that can be formed using all the letters of the word MONDAY, with the condition that no letter is repeating, is the number of permutation of 6 letters taken 4 at a time.
The formula to calculate the permutation is,
P n r = n ! ( n − r ) !
Where,n is the number of objects taken r at a time.
Substitute 6 for n and 4 for r in the above formula.
P 6 4 = 6 ! ( 6 − 4 ) ! = 6 ! 2 !
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
n ! = n ( n − 1 ) ! n ! = n ( n − 1 ) ( n − 2 ) ! [ n ≥ 2 ] ⋮
The permutation becomes,
P 6 4 = 6 × 5 × 4 × 3 × 2 ! 2 ! = 6 × 5 × 4 × 3 = 360
Thus, the total number of 4-letter words formed from the word MONDAY is 360.
(ii)
The number of words, with or without meaning that can be formed using all the letters of the word MONDAY, with the condition that no letter is repeating, is the number of permutation of 6 letters taken all at a time.
The formula to calculate the permutation is,
P n r = n ! ( n − r ) !
Where,n is the number of objects taken r at a time.
Substitute 6 for n and 6 for r in the above formula.
P 6 6 = 6 ! ( 6 − 6 ) ! = 6 ! 0 ! = 6 ! 1 = 6 !
The formula to calculate n ! is defined as,
n ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) × n
The factorial of 6 is,
P 6 6 = 6 × 5 × 4 × 3 × 2 × 1 = 720
Thus, the total numbers of words that can be formed with all letters of MONDAY are 720.
(iii)
The word MONDAY contains 2 vowels O and A. Since the first letter is a vowel and there are 2 vowels in the word, thus the number of ways in which the first place is occupied by a vowel is 2. Now, there are 5 places left to make a 6-letter word. Since, the repetition of letters is not allowed, so there are 5 letters left to occupy the 5 places.
It is the number of permutation of 5 letters taken all at a time.
The formula to calculate the permutation is,
P n r = n ! ( n − r ) !
Where,n is the number of objects taken r at a time.
Substitute 5 for n and 5 for r in the above formula.
P 5 5 = 5 ! ( 5 − 5 ) ! = 5 ! 0 ! = 5 ! 1 = 5 !
The formula to calculate n ! is defined as,
n ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) × n
Thus the factorial of 5 is,
P 5 5 = 5 × 4 × 3 × 2 × 1 = 120
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then total number of ways is m × n .
The number of ways that all letters are used but first letter is a vowel,
2 × 120 = 240
Thus, the total number of words that can be formed is 240.
The word MONDAY contains 6 different letters.
(i)
The number of 4-letter words, with or without meaning that can be formed using all the letters of the word MONDAY, with the condition that no letter is repeating, is the number of permutation of 6 letters taken 4 at a time.
The formula to calculate the permutation is,
Where,n is the number of objects taken r at a time.
Substitute 6 for n and 4 for r in the above formula.
Cancel the common factors by factorizing the bigger term to the factorial.
The formula to calculate the factors of a factorial in terms of factorial itself is,
The permutation becomes,
Thus, the total number of 4-letter words formed from the word MONDAY is 360.
(ii)
The number of words, with or without meaning that can be formed using all the letters of the word MONDAY, with the condition that no letter is repeating, is the number of permutation of 6 letters taken all at a time.
The formula to calculate the permutation is,
Where,n is the number of objects taken r at a time.
Substitute 6 for n and 6 for r in the above formula.
The formula to calculate
The factorial of 6 is,
Thus, the total numbers of words that can be formed with all letters of MONDAY are 720.
(iii)
The word MONDAY contains 2 vowels O and A. Since the first letter is a vowel and there are 2 vowels in the word, thus the number of ways in which the first place is occupied by a vowel is 2. Now, there are 5 places left to make a 6-letter word. Since, the repetition of letters is not allowed, so there are 5 letters left to occupy the 5 places.
It is the number of permutation of 5 letters taken all at a time.
The formula to calculate the permutation is,
Where,n is the number of objects taken r at a time.
Substitute 5 for n and 5 for r in the above formula.
The formula to calculate
Thus the factorial of 5 is,
By multiplication principle which states that if an event can occur in m different ways and follows another event that can occur in n different ways, then total number of ways is
The number of ways that all letters are used but first letter is a vowel,
Thus, the total number of words that can be formed is 240.
