How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
The word EQUATION contains 8 different letters from which 5 are vowels and 3 are consonants i.e. E, U, A, I and O are vowels and Q, T and N are consonants. When all the vowels occur together, then it is considered to be 1 letter and when the consonants occur together then they are also considered as 1 letter. Then total letters will be 2. Therefore, the arrangement of the word is the number of permutation of 2 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 2 for n and 2 for r in the above equation.
P 2 2 = 2 ! ( 2 − 2 ) ! = 2 ! 0 ! = 2 ! 1 = 2 !
The formula to calculate n ! is defined as,
n ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) × n
Thus the factorial of 2 is,
P 2 2 = 2 × 1 = 2
Thus, the total numbers of words that can be formed are 2.
Since there are 5 vowels, thus they are also being arranged in themselves. The number of arrangements of vowels is the combination of 5 vowels 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 equation.
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
Thus, the total numbers of words that can be formed are 120.
Since there are 3 consonants, thus they are also being arranged in themselves. The number of arrangements of vowels is the combination of 3 consonants 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 3 for n and 3 for r in the above equation.
P 3 3 = 3 ! ( 3 − 3 ) ! = 3 ! 0 ! = 3 ! 1 = 3 !
The formula to calculate n ! is defined as,
n ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) × n
Thus the factorial of 3 is,
P 3 3 = 3 × 2 × 1 = 6
Thus, the total numbers of words that can be formed are 6.
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, the number of ways that the words are selected is 2 × 120 × 6 = 1440 .
The word EQUATION contains 8 different letters from which 5 are vowels and 3 are consonants i.e. E, U, A, I and O are vowels and Q, T and N are consonants. When all the vowels occur together, then it is considered to be 1 letter and when the consonants occur together then they are also considered as 1 letter. Then total letters will be 2. Therefore, the arrangement of the word is the number of permutation of 2 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 2 for n and 2 for r in the above equation.
The formula to calculate
Thus the factorial of 2 is,
Thus, the total numbers of words that can be formed are 2.
Since there are 5 vowels, thus they are also being arranged in themselves. The number of arrangements of vowels is the combination of 5 vowels 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 equation.
The formula to calculate
Thus the factorial of 5 is,
Thus, the total numbers of words that can be formed are 120.
Since there are 3 consonants, thus they are also being arranged in themselves. The number of arrangements of vowels is the combination of 3 consonants 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 3 for n and 3 for r in the above equation.
The formula to calculate
Thus the factorial of 3 is,
Thus, the total numbers of words that can be formed are 6.
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, the number of ways that the words are selected is
