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Math  /  Discrete

QuestionConsider the arrangement of all letters of word MOTALE. Then
One/More correct answer(s) A. Number of arrangements in which vowels and consonants are alternate is 2.3!32.3!3 ! B. Number of arrangements in which vowels are alternate is 3!33!3 ! C. Number of arrangements in which consonants come before vowels is 3 ! 3 ! D. Number of arrangements in which vowels are in alphabetical order is 6!3!\frac{6!}{3!}

Studdy Solution

STEP 1

1. The word "MOTALE" consists of 6 letters: 3 vowels (O, A, E) and 3 consonants (M, T, L).
2. We need to evaluate the correctness of each given option based on the arrangement of these letters.

STEP 2

1. Analyze the arrangement where vowels and consonants alternate.
2. Analyze the arrangement where vowels are alternate.
3. Analyze the arrangement where consonants come before vowels.
4. Analyze the arrangement where vowels are in alphabetical order.

STEP 3

To have vowels and consonants alternate, we can start with either a vowel or a consonant.
- If we start with a vowel, the sequence is V-C-V-C-V-C. - If we start with a consonant, the sequence is C-V-C-V-C-V.
There are 2 possible starting patterns.

STEP 4

Calculate the number of ways to arrange the vowels (O, A, E) and consonants (M, T, L).
- Vowels can be arranged in 3!3! ways. - Consonants can be arranged in 3!3! ways.
Thus, the total number of arrangements is 2×3!×3!2 \times 3! \times 3!.

STEP 5

For vowels to be alternate, the arrangement is similar to STEP_1.
- We already calculated this in STEP_1, so the answer is 2×3!×3!2 \times 3! \times 3!.

STEP 6

For consonants to come before vowels, arrange the consonants first and then the vowels.
- Consonants can be arranged in 3!3! ways. - Vowels can be arranged in 3!3! ways.
Thus, the total number of arrangements is 3!×3!3! \times 3!.

STEP 7

For vowels to be in alphabetical order, fix the order of vowels as A, E, O.
- The number of ways to arrange the consonants is 3!3!.
Thus, the total number of arrangements is 3!3!.
Now, let's evaluate each option:
A. Number of arrangements in which vowels and consonants are alternate is 2×3!×3!2 \times 3! \times 3!. This is correct. B. Number of arrangements in which vowels are alternate is 2×3!×3!2 \times 3! \times 3!. This is correct. C. Number of arrangements in which consonants come before vowels is 3!×3!3! \times 3!. This is correct. D. Number of arrangements in which vowels are in alphabetical order is 3!3!. This is incorrect; it should be 3!3!, not 6!3!\frac{6!}{3!}.
Correct answers: A, B, C.

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