A new word is to be formed by randomly rearranging the letters of the

Question

A new word is to be formed by randomly rearranging the letters of the word ALGEBRA. What is the probability that the new word has consonants occupying only the positions currently occupied by consonants in the word ALGEBRA?

(A) 2/120
(B) 1/24
(C) 1/6
(D) 2/105
(E) 1/35

(A) 2/120
(B) 1/24
(C) 1/6
(D) 2/105
(E) 1/35

eswarchethu135 · Answer

Total number of ways of arranging the word ALGEBRA is  \frac{7!}{2!}

Now the consonants occupy the same position as in the word ALGEBRA. So consonants will be _LG_BR_.

The 3 vowels in the 3 blanks can be arranged in  \frac{3!}{2!}  ways.

The consonants can be arranged in those positions in 4!

Probability =  \frac{(\frac{3!}{2!}) * 4!}{\frac{7!}{2!}}

=  \frac{3*4!}{3*4*5*6*7}

=  \frac{1}{35}

OPTION:  E

Archit3110 · Answer

ALGEBRA ;
total Probability ; 7!/2!
Consonants positions can be taken in 4! ways
and vowels 3!/2! ways arrangement
so 
P ; 4!*3!/2! / 7!/2!
solve we get
4!*3/7! ; 1/35 
IMO E

Albs · Answer

Archit3110  Why did you divide total probability by 2?

IanStewart · Answer

We have fewer vowels, so it's slightly easier to answer the equivalent question "what is the probability the vowels stay in the same place". If we choose where to put the vowels in our new word, we have seven positions to choose from (first letter, second letter, etc) and we're choosing three of those places. We can choose three things from seven in 7C3 = (7)(6)(5)/3! = 35 ways. In only one of those ways have the vowels remained in their previous locations, so the probability the vowels are where vowels used to be (and thus that consonants are where consonants used to be) is 1/35.

arronjam · Answer

Fill in the vowels first then consonants positions

Vowels positions can be filled in 3 ways

there are 4 consonants they can be filled in 4! ways

Multiplying 3 by 24 gives 72 ways to arrange the letters given the restrictions. Total ways to arrange the letters ALGEBRA is 7!/2!

72/(7! /2!) gives 1/35

OhsostudiousMJ · Answer

Probability method of solving:

Question requires the consonants occupying the same positions (they can interchange - a mistake I did at first). And if the consonants are at their positions, then the vowels would also need to be in their respective positions (and they can interchange as well).
There are initially 3 vowels and 4 consonants.

Hence position-wise probability =  \frac{3}{7}*\frac{4}{6}*\frac{3}{5}*\frac{2}{4}*\frac{2}{3}*\frac{1}{2}*1 = \frac{1}{35}

Answer  E .

hudacse6 · Answer

any other way

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