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
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.
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A word has 8 consonants and 3 ...
Updated On: 27-06-2022
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Answer
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