Question

Out of 7 Consonants and 4 vowels, words are to be formed by involving 3 consonants and 2 vowels. The number of such words are formed is:
A. 25200
B. 22500
C. 10080
D. 5040

Answer 1

Hint- In this question we use the theory of permutation and combination. We have to choose out of 7 Consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed. For example, in this case, how we choose two vowels out of four vowels and this can be done in ${}^{\text{4}}{{\text{C}}_2}$ =6 ways.

Complete step-by-step answer:
As we know,
${}^{\text{n}}{{\text{C}}_{\text{r}}}{\text{ = }}\dfrac{{{\text{n}}!}}{{{\text{[r}}!{\text{(n - r)}}!{\text{]}}}}$
Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4) is calculated using the formula-
${}^{\text{n}}{{\text{C}}_{\text{r}}}{\text{ = }}\dfrac{{{\text{n}}!}}{{{\text{[r}}!{\text{(n - r)}}!{\text{]}}}}$
Now,
${}^{\text{7}}{{\text{C}}_{\text{3}}} \times {}^{\text{4}}{{\text{C}}_{\text{2}}}$= $\dfrac{{{\text{7!}}}}{{{\text{(7 - 3)!3!}}}} \times \dfrac{{{\text{4!}}}}{{{\text{(4 - 2)!2!}}}}$
= $\dfrac{{7 \times 6 \times 5}}{{3 \times 2 \times 1}} \times \dfrac{{4 \times 3}}{{2 \times 1}}$
= 210
Number of groups, each having 3 consonants and 2 vowels =210.
Each group contains 5 letters.
Number of ways of arranging 5 letters among themselves =5!
= 5×4×3×2×1
= 120
∴ Required number of ways = (210×120) =25200.
Hence, the answer is 25200.
So, option (A) is the correct answer.

Note- Permutations and combinations, the various ways in which objects from a set may be selected, generally without replacement, to form subsets. This selection of subsets is called a permutation when the order of selection is a factor, a combination when order is not a factor. Thus, if we want to figure out how many combinations we have of n objects then r at a time, we just create all the permutations and then divide by r! variant.