Four letters are taken at random from the word AUSTRALIA without repla

Question

From the word AUSTRALIA, a letter is taken at random and put back. Then a second letter is taken and put back. This process is repeated for four letters. What would be the possibility that two vowels and two consonants have been chosen?

A.  \frac{20^2}{9^4}

B.  \frac{2*20^2}{9^4}

C.  \frac{3*20^2}{9^4}

D.  \frac{4*20^2}{9^4}

E.  \frac{6*20^2}{9^4}

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chetan2u · Answer

Total 9 alphabets, out of which 5 are vowels and 4 are consonants. 
Way to choose 4 out of 9 alphabets with replacement = 9*9*9*9=9^4

We can choose two vowels and two consonants in 5*5*4*4=20^2
But the ways to choose 4 alphabets ( 2V and 2C)=4!/2!2!=6
Total ways 6*20^2

Probability = \frac{6*20^2}{9^4}

E

Harshjha001 · Answer

IMO A 
5/9*5/9*4/9*4/9 = sqr(20)/9*9*9*9

CEdward · Answer

The question is asking: P(exactly 2 vowels) = ?

Suppose we picked out the vowels and consonants in this order:

VVCC

5/9 x 5/9 x 4/9 x 4/9 = (5^2 x 4^2)/9^4

But, VVCC is only one possible order. Here are the others:

VCVC
CVCV
CCVV
VCCV
CVVC

So we must multiply 5^2 x 4^2/9^4 by 6 --> 6 x (5^2x4^2)/9^4

ahujaparth10 · Answer

Hey  Bunuel

We are asked to find out the possibilities of picking two Vowels & two consonants. When we do 5 x 5 x 4 x 4 - We find out the possible number of ways 2 vowels & 2 consonants are picked in a particular order (V-V-C-C). However, why is order important here & why are we multiplying by 6? I would consider VCVC to be the same as VVCC as it's still picking 2 vowels & 2 consonants.

Bunuel · Answer

The reason we multiply by 6 is that there are 6 different sequences (ways) in which two vowels and two consonants can be chosen:

1. V-V-C-C
2. V-C-V-C
3. V-C-C-V
4. C-V-V-C
5. C-V-C-V
6. C-C-V-V

The probability of each of these sequences happening is 5/9 * 5/9 * 4/9 * 4/9. By accounting for all 6 sequences (6 is essentially the number of ways 4 letters VVCC can be arranged: 4!/(2!2!) = 6), we capture the full probability of choosing two vowels and two consonants.

bumpbot · Answer

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Four letters are taken at random from the word AUSTRALIA without repla

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Four letters are taken at random from the word AUSTRALIA without repla

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