Read the information given below carefully and answer the question that follows:
(A) The different ways in which the alphabets of the word BAKERY can be arranged is 720
(B) The number of ways in which the alphabets of the word MACHINE can be arranged so that the vowels will occupy only the odd positions is 576
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Both (A) and (B)
Statement (A) Verification
The word BAKERY has 6 distinct letters (B, A, K, E, R, Y).
The number of ways to arrange $n$ distinct objects is given by $n!$ (n-factorial).
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$
Conclusion: Statement (A) is True.
Statement (B) Verification
The word MACHINE has 7 letters:
- Vowels: A, I, E (3 total)
- Consonants: M, C, H, N (4 total)
There are 7 positions to fill: 1, 2, 3, 4, 5, 6, 7.
The odd positions are: 1, 3, 5, 7 (4 positions).
- Arrange the Vowels: We need to place 3 vowels into 4 available odd positions. Number of ways = $^4P_3 = 4 \times 3 \times 2 = 24$ ways.
- Arrange the Consonants: There are 4 positions remaining (3 even positions + 1 leftover odd position) for the 4 consonants. Number of ways = $4! = 4 \times 3 \times 2 \times 1 = 24$ ways.
Total ways = $24 \times 24 = 576$.
Conclusion: Statement (B) is True.
Final Answer: Both statements are mathematically correct.
Result: Both (A) and (B)