Words are formed using the letters of the word 'POTATO' Let 'a' denotes the no. of words when all
the vowels are together. 'b' denotes the no. of words when vowels as well as consonants are
separated and 'c' denotes the no. of words which begin and end in vowels. Which of the following is
(are) correct?
(A) a >>
(B) ac4b
(C) a b c 90 (D) 3b 2c
Answers
Answer:
anwer is b because it can make the word
Answer:
The correct answer is (A) and (C).
Step-by-step explanation:
To solve this problem, we can use basic principles of permutation and combination.
a) When all vowels are together:
There are 3 vowels in the word 'POTATO' (O occurs twice). We can treat these 3 vowels as a single letter and arrange them among themselves in 3! = 6 ways. Now we have 4 letters (P, T, T, and the group of vowels) that can be arranged in 4! = 24 ways. Therefore, the total number of words in which all vowels are together is 6 x 24 = 144.
Hence, statement (A) is correct.
b) When vowels and consonants are separated:
We can treat the 2 T's as distinct letters and arrange all 6 letters in 6! = 720 ways. However, we need to subtract the cases where all vowels are together because we have already counted them in part (a). From part (a), we know that there are 144 words where all vowels are together. There are 3 places where the group of vowels can be inserted into the 4 consonants: before the first consonant, between the first and second consonants, and after the last consonant. Each of these 3 cases can be arranged in 3! = 6 ways. Therefore, the total number of words in which vowels and consonants are separated is 720 - 3 x 144 x 6 = 72.
Hence, statement (B) is incorrect.
c) Words that begin and end in vowels:
There are 2 vowels (O occurs twice) in the word 'POTATO'. We can treat these 2 vowels as distinct and arrange them in 2! = 2 ways at the beginning and end of the word. For the middle 4 letters, we have 4! = 24 ways to arrange them. Therefore, the total number of words that begin and end in vowels is 2 x 24 = 48.
Hence, statement (C) is correct.
Therefore, the correct answer is (A) and (C).
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