Answer:
1440
Explanation:
The answer is not as simple as you might think. You can't just multiply 5 by 4 by 3 by 2 and get 120. That would be too easy. You have to consider the order of the positions and the gender of the candidates. For example, you can't have a woman as president and another woman as vice president, because that would violate the rule of 2 women and 2 men. You also can't have the same person as president and secretary, because that would be cheating.
This can be solved using the combination formula. But before we do that, let's make some funny assumptions to spice things up. Let's assume that:
- The president must be a woman, because women are better leaders than men (just kidding).
- The vice president must be a man, because men are better at following orders than women (again, just kidding, please don't cancel me).
- The secretary must be a woman, because women have better handwriting than men (OK, this one might be true).
- The treasurer must be a man, because men are better at handling money than women (OK, this one is definitely not true).
Now that we have these hilarious and totally not gender related criteria, we can use the combination formula to find out how many ways the executive body can be formed. The formula is: n!/(n-r)!
where n is the total number of things and r is the number of things you want to arrange. For example, if you have 5 things and you want to arrange 3 of them, the formula is 5!/(5-3)! = 5!/2! = (5*4*3*2*1)/(2*1) = 60.
But wait, there's more! You also have to use another formula called the combination formula, which tells you how many ways you can choose a certain number of things from a larger group without caring about the order. The formula is n!/(r!(n-r)!), where n is the total number of things and r is the number of things you want to choose. For example, if you have 5 things and you want to choose 3 of them, the formula is 5!/(3!(5-3)!) = (5*4*3*2*1)/(3*2*1)(2*1) = 10.
So how do these formulas help us with our problem? Well, first we have to choose 2 women out of 4, which can be done in 4!/(2!(4-2)!) = 6 ways. Then we have to choose 2 men out of 5, which can be done in 5!/(2!(5-2)!) = 10 ways. Then we have to arrange these 4 people in the 4 positions, which can be done in 4!/(4-4)! = 24 ways. Finally, we have to multiply these numbers together to get the total number of ways: 6 * 10 * 24 = 1440.
That's right, there are 1440 possible ways to form the executive body with these conditions. Isn't that amazing?