Multiple choice

Given below are two statements: Statement I: A committee of 4 can be made out of 5 men and 3 women containing at least one woman in 65 ways. Statement II: The number of words which can be formed using letters of the word ARRANGE so that vowels always occupy even place is 36. In light of the above statements, choose the correct answer from the options given below.

  1. Both Statement I and Statement II are true.

  2. Both Statement I and Statement II are false.

  3. Statement I is true, but Statement II is false.

  4. Statement I is false, but Statement II is true.
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A Correct answer
Explanation

Statement I: Total ways to choose 4 from 8 is 8C4 = 70. Ways with no women (choosing 4 from 5 men) is 5C4 = 5. Ways with at least one woman = 70 - 5 = 65. True. Statement II: ARRANGE has 7 letters (A:2, R:2, N:1, G:1, E:1). Vowels are A, A, E (3). Even positions are 2, 4, 6 (3). We place 3 vowels in 3 spots: 3! / 2! = 3 ways. Remaining 4 letters (R, R, N, G) in 4 spots: 4! / 2! = 12 ways. Total = 3 * 12 = 36. True.

AI explanation

For Statement I, the number of ways to form the committee is the sum of combinations with 1, 2, and 3 women, calculated as (5C3 x 3C1) + (5C2 x 3C2) + (5C1 x 3C3), which equals (10 x 3) + (10 x 3) + (5 x 1) = 65, making it true. For Statement II, the word ARRANGE has 3 vowels (A, A, E) and 4 consonants, and placing vowels in the 3 even places gives 3 factorial divided by 2 factorial arrangements for vowels, while the consonants give 4 factorial divided by 2 factorial arrangements. Multiplying these gives (3 x 2 x 1 / 2) x 24 = 36, making Statement II true as well, which means both statements are true.

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