The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?
The English alphabet has 5 vowels and 21 consonants. Since 2 vowels are selected from 5 vowels, thus the number of ways that the vowels are selected is the number of combination of 5 vowels taken 2 at a time.
The formula for combination is,
C n r = n ! ( n − r ) ! r !
Substitute 5 for n and 2 for r in the above formula,
C 5 2 = 5 ! ( 5 − 2 ) ! 2 ! = 5 ! 3 ! 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 combination becomes,
C 5 2 = 5 × 4 × 3 ! 3 ! 2 × 1 = 5 × 4 2 × 1 = 10
Thus the number of ways that the vowels are selected is 10.
Since 2 consonants are selected from 21 consonants, thus the number of ways that the consonants are selected is the number of combination of 21 consonants taken 2 at a time.
The formula for combination is,
C n r = n ! ( n − r ) ! r !
Substitute 21 for n and 2 for r in the above formula,
C 21 2 = 21 ! ( 21 − 2 ) ! 2 ! = 21 ! 19 ! 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 combination becomes,
C 21 2 = 21 × 20 × 19 ! 19 ! 2 × 1 = 21 × 20 2 × 1 = 210
Thus the number of ways that the consonants are selected is 210.
Each combination of 2 vowels and 2 consonants contains 4 letters in total which are also being arranged among them. The number of ways that the 4 letters are arranged is 4 ! .
The formula to calculate n ! is defined as,
n ! = 1 × 2 × 3 × ⋯ × ( n − 1 ) × n
Thus the factorial of 4 is,
4 ! = 4 × 3 × 2 × 1 = 24
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 of selecting 2 vowels and 2 consonants are 10 × 210 × 24 = 50400 .
The English alphabet has 5 vowels and 21 consonants. Since 2 vowels are selected from 5 vowels, thus the number of ways that the vowels are selected is the number of combination of 5 vowels taken 2 at a time.
The formula for combination is,
Substitute 5 for n and 2 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 combination becomes,
Thus the number of ways that the vowels are selected is 10.
Since 2 consonants are selected from 21 consonants, thus the number of ways that the consonants are selected is the number of combination of 21 consonants taken 2 at a time.
The formula for combination is,
Substitute 21 for n and 2 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 combination becomes,
Thus the number of ways that the consonants are selected is 210.
Each combination of 2 vowels and 2 consonants contains 4 letters in total which are also being arranged among them. The number of ways that the 4 letters are arranged is
The formula to calculate
Thus the factorial of 4 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, the number of ways of selecting 2 vowels and 2 consonants are
