All possible 5 lettered words are formed using all the letters of the word "COCOA" and a word from it is randomly selected. The chance that any two alike letters are not adjacent in the selected word, is

Question

All possible 5 lettered words are formed using all the letters of the word "COCOA" and a word from it is randomly selected. The chance that any two alike letters are not adjacent in the selected word, is (A) (1/5) (B) (2/5) (C) (1/3) (D) (2/3)

Tardigrade Admin · Accepted Answer

n(S)=(5 !/2 ! 2 !)=30 E: required event n ( E )=30-( n ( A ∪ B )) where n(A) is number of words when C's togather etc.

n(A ∪ B)=n(A)+n(B)-n(A ∩ B) =((4 !/2 !))+(4 !/2 !)-3 !=24-6=18=12

P(E)=(12/30)=(2/5)

Q. All possible 5 lettered words are formed using all the letters of the word "COCOA" and a word from it is randomly selected. The chance that any two alike letters are not adjacent in the selected word, is

$\frac{1}{5}$

$\frac{2}{5}$

$\frac{1}{3}$

$\frac{2}{3}$

$n (S) = \frac{5 !}{2 ! 2 !} = 30$
$E$ : required event
$n (E) = 30 - (n (A \cup B))$
where $n (A)$ is number of words when C's togather etc.

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$
$= (\frac{4 !}{2 !}) + \frac{4 !}{2 !} - 3! = 24 - 6 = 18 = 12$

$P (E) = \frac{12}{30} = \frac{2}{5}$