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Total no. of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to ?
Ans. could be 1]60 2]120 3]7200 4]none I solved it like as these 2 events don't relate to each other hence calculate them separately. So 4C2 x 3C5 = 60 . Is it right?
asked Jan 22, 2014 at 17:57
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Choose the consonants: ${5 \choose 3} = 10$.
Choose the vowels: ${4 \choose 2} = 6$.
Choose what order they appear in: $5! = 120$.
That gives $7200$.
answered Jan 22, 2014 at 18:02
JohnJohn
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Total number of words formed by 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is equal to
60
120
7200
none of these
7200
2 out of 4 vowels can be chosen in 4C2 ways and 3 out of 5 consonants can be chosen in 5C3 ways.
Thus,
there are \[\left[ C_2 \times {}^5 {C^4}_3 \right]\] groups, each containing 2 vowels and 3 consonants.
Each group contains 5 letters that can be arranged in 5! ways.
∴ Required number of words =\[\left[ {}^4 C_2 \times {}^5 C_3 \right] \times 5! = 60 \times 120 = 7200\]
Total no. of words formed by using 2 vowels and 3 consonants taken from 4 vowels and 5 constants in equal to:
A. 60
B. 120
C. 720
D. None of these
Answer
Verified
Hint: This problem deals with permutations and combinations. But here a simple concept is used. Although this problem deals with combinations only. Here factorial of any number is the product of that number and all the numbers less than that number till 1.
$ \Rightarrow n! = n[n - 1][n - 2].......1$
The no. of combinations of $n$ objects taken $r$ at a time is determined by the formula which is used:
$ \Rightarrow {}^n{c_r} = \dfrac{{n!}}{{\left[ {n - r} \right]!r!}}$
Complete
step-by-step answer:
Given that there are 4 vowels and 5 consonants.
We are asked to find the total no. of words formed by the words taken 2 vowels from the given 4 vowels and 3 consonants from the given 5 consonants.
So we have to select 2 vowels from the given 4 vowels.
The no. of ways we can select 2 vowels from the given 4 vowels is given by:
$ \Rightarrow {}^4{c_2} = 6$
We have to select 3 consonants from the given 5 consonants.
The no. of ways we can select 3
consonants from the given 5 consonants is given by:
$ \Rightarrow {}^5{c_3} = 10$
So now we have 2 vowels and 3 consonants, which means that we have 5 letters in total.
The no. of ways in which these 5 letters can be arranged is given by:
$ \Rightarrow 5! = 120$
Now in these 5 letters, the 2 vowels can be arranged in ${}^4{c_2}$ ways and the 3 consonants can be arranged in ${}^5{c_3}$ ways, which is given by:
$ \Rightarrow 120 \times {}^4{c_2} \times {}^5{c_3} = 120 \times 6
\times 10$
$ \Rightarrow 7200$
Final Answer: Total no. of words formed by using 2 vowels and 3 consonants taken from 4 vowels and 5 constants in equal to 7200.
Note:
While solving this problem please note that while considering the total no. of words which can be formed with using 2 vowels and 3 consonants taken from 4 vowels and 5 constants, here though the selected 5 letters can be arranged in 120 ways, but again in these 5 letters again the 2 vowels from
the selected 4 vowels can be arranged in 6 ways and the 3 consonants from the selected 5 consonants can be arranged in 10 ways, hence multiplied with these.
the question is that total number of words formed by 3 vowels and 3 consonants taken from five vowels and consonants is equal to that we need to find out so number of ways for consonant using is equal to total number is 5 and we have to choose three it will be 5C 3 night will be equal to 5 factorial by 3 factorial in 25 - 3 factorial because we are using the formula NCR is equal to an ethereal by architect tutorial in 2 and minus are real this will be equal to 5 into 4 into 3 factorial by 3 factorial into two factorial 3
and we are left with 5 into 4 by 2 is equal to 10 on the same pattern number of ways of mobile is equal to 5 total out of that we have to choose three so it will again be 10 and number of ways the words the a is equal to number of 3 + number of consonant 3 and factorial of that so it will be equal to 6 factorial which is equal to 720 from this we can find therefore total number of ways is equal to 10 into 10 into 720 which is equal to 70
therefore we can write our answer as answer is option number 3 which is 72000 I hope you to destination thank you