Question

How many words with or without meaning taking 3 consonants and 2 vowels can be formed using 5 consonants and 4 vowels?

Answer 1

Hint – In this question first of all select 3 consonants out of 5 given consonants and 2 vowels out of 4 given vowels using combination rule (i.e. to select r objects out of n objects we use ${}^n{C_r}$), later on in the solution arrange these selected consonants and vowels so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
5 consonants and 4 vowels.
Now we have to make words with or without meaning using these given consonants and vowels by taking 3 consonants and 2 vowels out of given consonants and vowels.
So the number of ways to choose 3 consonants out of 5 given consonants = ${}^5{C_3}$
And the number of ways to choose 2 vowels out of 4 vowels = ${}^4{C_2}$
So total selection of the letters (i.e. 3 consonants and 2 vowels) is (3 + 2) = 5 letters.
So the number of ways to arrange them is (5!).
So the total number of words with or without meaning using 3 consonants and 2 vowels out of given consonants and vowels are the multiplication of all of the above calculated values so we have,
Total number of words with or without meaning = ${}^5{C_3} \times {}^4{C_2} \times 5!$
Now as we know that \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] so use this property in the above equation we have,
Total number of words with or without meaning = $\dfrac{{5!}}{{3!\left( {5 - 3} \right)!}} \times \dfrac{{4!}}{{2!\left( {4 - 2} \right)!}} \times 5!$
Now simplify this we have,
Total number of words with or without meaning = $\dfrac{{5!}}{{3!.2!}} \times \dfrac{{4!}}{{2!.2!}} \times 5!$
$ = \dfrac{{5.4.3!}}{{3!.2.1}} \times \dfrac{{4.3.2!}}{{2.1.2!}} \times 5 \times 4 \times 3 \times 2 \times 1$
$ = 10 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$ = 10 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 7200$ Words.
So this is the required answer.

Note – In such types of questions the key concept we have to remember is the formula of the combinations which is given as \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] so use this formula to get on the right track while simplifying the combination equation as above, we will get the required answer.