How can 7 consonants and 4 vowels words of three consonants and two vowels can be formed?
Out of 7 consonants and 4 vowels, how many words can be formed such that it contains 3 consonants and 2 vowels ?
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Answer (Detailed Solution Below)Option 3 : 25200 Concept: The number of ways to select r things out of n given things wherein r ≤ n is given by: \({\;^n}{C_r} = \frac{{n!}}{{r!\; × \left( {n - r} \right)!}}\) Calculation: Given: There are 7 consonants and 4 vowels Here, we have to find how many word can be formed such that it has 3 consonants and 2 vowels. No. of ways to select 2 vowels out of 4 vowels = \({\;^4}{C_2} \) No. of ways to select 3 consonants out of 7 consonants = \({\;^7}{C_3} \) ∴ No. of words that can be formed which contains 3 consonants and 2 vowels = \({\;^4}{C_2} \) × \({\;^7}{C_3} \) As we know that, \({\;^n}{C_r} = \frac{{n!}}{{r!\; × \left( {n - r} \right)!}}\) ⇒ \({\;^4}{C_2} \) × \({\;^7}{C_3} \) = 6 × 35 = 210 The no. of ways to arrange words containing 3 consonants and 2 vowels = 210 × 5! = 25200 Hence, option C is the correct answer. Stay updated with the Mathematics questions & answers with Testbook. Know more about Permutations and Combinations and ace the concept of Combination Function and its properties. Answer Verified Hint: First of all, find the number of ways of choosing the vowels and consonants separately. Then, find the ways to choose both of them. And finally, find the number of ways of arranging them by multiplying it with 5!. Complete step-by-step answer: Note: Students generally get confused between permutations and combinations as combination means only choosing while permutation means first choosing and then arranging. First we find 5 letters out of given letters and then arrange them using the formula 5! = $ 5 \times 4 \times 3 \times 2 \times 1$. 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. 25200B. 22500C. 10080D. 5040Answer Verified 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: 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. How can 7 constants and 4 vowels words of three consonants and two vowels be formed?Number of groups, each having 3 consonants and 2 vowels =210.
How many words can be formed using 3 consonants and 2 vowels?= 210. Number of groups, each having 3 consonants and 2 vowels = 210.
How many words of 4 consonants and 3 vowels can be formed?∴ Required number of words =12C4∗4C3∗7! =9979200.
How many words of 3 consonants and 2 vowels can be formed from 5 consonants and 4 vowels?From 5 consonants and 4 vowels, how many words can be formed by using 3 consonants and 2 vowels. A. 9440.
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