(2) is pretty trivial. Assume there are k members, then they can be permuted k! ways so k!=720 -> k=6
(1) I'm not sure of yet
One of my biggest challenges in math is solving word problems. A word problem is given. I have never seen it before but it is not hard to solve. However, I cannot, for the life of me, reason the problem out or create an equation leading to the right answer.
I am not talking about word problems that I've recently been taught to solve. I am talking about correctly answering word problems usually given on standardized exams. On the other hand, there are people who did not major in math who can reason their way through a math word problem and without creating the needed equation, can successfully answer the applications or word problems.
I find, for example, probability word problems to be a bit fuzzy. In fact, I find probability word problems far more complicated than those usually found in algebra 2, college algebra and precalculus trigonometry textbooks.
Geometry word problems at the high school level are not too bad. Below are two word problems that someone like me, who has been playing with math textbooks for many years, should be able to solve with ease. Honestly, I don't know where to start.
Two Sample Word Problems
(1) 40 boys and 28 girls stand in a circle, hand in hand, all facing inwards. Exactly 18 boys give their right hand to a girl. How many boys give their left hand to a girl?
(2) The Coughlin family discovered they can stand in a row for their family portrait 720 different ways. How many members are in the Coughlin family?
Please, explain how you would reason your way through the questions.
For (2) you must first decide what type of math are you going to use to arrive at your answer, that is the real trick, how many formulas can you memorize or semi memorize meaning you have a good idea where to start, lets start with an educated guess. Do we use Purmutations, combinations, will factorals be involved, do we just draw it out by long hand on paper and forget the math for the initial guesses. If there are 5 people in the family and one person gets in line, that leaves 4 more people to get in second place, then 3 more people to get into third place and so on so this is a Factoral. What Factoral equals 720 possibilities. In this case 5x4x3x2x1=120 Of course you need to reverse engineer the problem, check this out. Factorial number system - Wikipedia, the free encyclopedia
For (1) you have to assume that boys and girls are standing in equal groups at the same ratio of total boys and girls, 40:28, subgroup 20:14, subgroup again 10:7. You also have to think that boys and girls are split up evenly within the subgroups which could be B G B G B G B B G B B G B G B G B which is balanced. This shows 4 even subgroups but does it fit your pattern. 18/4 = not an even integer, so not all subgroups are evenly aligned. Split the groups into two, 20:14, now 9 boys gave their right hand to a girl, there are only 10 girls total still unmentioned so 5 per group of 20:14 available to give a left hand, arrange everybody in that fashion and see what works out. Long hand math is long isnt it. Ratios are a problem with me too. But every ratio is a fraction, 40/28, 20/14, 10/7. Someone smarter than us will be along shortly.