Question

In how many ways can a vowel, a consonant and a digit be chosen out of 26 letters of the English alphabets and the 10 digits?

Answer 1

Hint: We write total number of consonants and vowels available in the alphabets list. Calculate number of ways to choose a vowel from list of vowels, number of ways to choose a consonant from list of consonants and number of ways to choose a digit from list of digits.
* In 26 English alphabet letters there are 5 vowels i.e. A, E, I, O and U and rest are consonants
* There are 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
* Formula of combination is given by\[^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\], where factorial is expanded by the formula \[n! = n \times (n - 1)! = n \times (n - 1) \times (n - 2)!.... = n \times (n - 1) \times (n - 2)....3 \times 2 \times 1\]

Complete step by step answer:
Formula of combination is given by\[^n{C_r} = \dfrac{{n!}}{{(n - r)!r!}}\], where factorial is expanded by the formula \[n! = n \times (n - 1)! = n \times (n - 1) \times (n - 2)!.... = n \times (n - 1) \times (n - 2)....3 \times 2 \times 1\]
If \[r = 1\]then\[^n{C_1} = \dfrac{{n!}}{{(n - 1)!1!}}\]
\[{ \Rightarrow ^n}{C_1} = \dfrac{{n(n - 1)!}}{{(n - 1)!}}\]
Cancel same terms from numerator and denominator
\[{ \Rightarrow ^n}{C_1} = n\] … (1)
We calculate number of ways to choose each vowel, a consonant and a digit separately.
A vowel:
Since there are 5 vowels i.e. A, E, I, O and U in total 26 letters of English alphabets
We have to choose 1 vowel from 5 vowels
We have total number of vowels to choose from as 5, so value of \[n = 5\]
We have to choose one vowel from the given vowels, so value of \[r = 1\]
Therefore, number of ways to choose is \[^5{C_1}\]
Using the equation (1) we get \[^5{C_1} = 5\] … (2)
A consonant:
Since there are 5 vowels i.e. A, E, I, O and U in total 26 letters of English alphabets, then number of consonants is \[26 - 5 = 21\]
We have to choose 1 consonant from 21 consonants
We have total number of consonants to choose from as 21, so value of \[n = 21\]
We have to choose one consonant from the given consonants, so value of \[r = 1\]
Therefore, number of ways to choose is \[^{21}{C_1}\]
Using the equation (1) we get \[^{21}{C_1} = 21\] … (3)
A digit:
Since there are 10 digits in number system
We have to choose 1 digit from 10 digits
We have total number of digits to choose from as 10, so value of \[n = 10\]
We have to choose one digit from the given digits, so value of \[r = 1\]
Therefore, number of ways to choose is \[^{10}{C_1}\]
Using the equation (1) we get \[^{10}{C_1} = 10\] … (4)
Total number of ways to choose a vowel, a consonant and a digit is given by multiplication of the number of ways to choose each of them.
Therefore, total number of ways to choose \[ = 5 \times 21 \times 10\]
On multiplying the values we get number of total ways to choose as 1050

\[\therefore \] Total number of ways to choose one vowel, one consonant and one digit is 1050.

Note: Alternative method:
We can directly solve for total number of ways without using combinations formula
Since we have 5 vowels, and we have to choose 1 vowel
Number of ways to choose a vowel \[ = 5\]
Since we have 21 consonants, and we have to choose 1 consonant
Number of ways to choose a consonant \[ = 21\]
Since we have 10 digits, and we have to choose 1 digit
Number of ways to choose a digit \[ = 10\]
\[ \Rightarrow \]Total number of ways to choose a vowel, a consonant and a digit \[ = 5 \times 21 \times 10\]
\[ \Rightarrow \]Total number of ways to choose a vowel, a consonant and a digit \[ = 1050\]
\[\therefore \] Total number of ways to choose one vowel, one consonant and one digit is 1050.