A Small Party

At the party the speaker made following 6 statements

[1] There were 9 men and children.

[2] There were 2 more women than children.

[3] The number of different man – woman couples possible was 24. ( If there were 10 men and 5 women at the party, then there would have been 10 x 5 or 50 man – woman couples possible.)

Of the three categories – men, women, and children – at the party:
[4] There were 4 people in one of the categories

[5] There were 6 people in one of the categories
[6] There were 8 people in one of the categories 

 Exactly one of the 6 statements made by speaker is false

Which of [1] through [6] is false?

Click here to see the solution.

Correct Answer : Statement (4) is false

Explanation:
Suppose [4] through [6] are all true. Then [1] is false, and [2] and [3] cannot be true. From [7], this situation is impossible.
So, from [7], one of [4] through [6] is false, and [1] through [3] are all true. Then from [1] and [2], there were 11 men and women. Then, from [1] and [2], there were 8 men and 3 women; or (B) there were 3 men and 8 women. Suppose A is correct. Then, from[2], there was 1 child. Then [4] and [5] are both false, contradicting [7]. So A is not correct. Then B is correct. Then, from [2], there were 6 children. Then [4] is false.