There are 5 freshmen and 6 sophomores in a class. We want to choose 3 students to represent the class in a competition. Find the number of combinations, where 2 freshmen and one sophomore are chosen.
Let us first find the number of ways to choose the 2 freshmen. Since the students are chosen "without replacement" and order doesn't matter (it doesn't matter which student is chosen first), we can use the number of combinations of 2 out of 5 possible. This gives us "5 Choose 2", which we can compute using a calculator, excel, or the combinations formula. Plugging in =Combin(5,2) into excel, we get 10. So there are 10 ways to pick the 2 freshmen. Now, we just use the same approach for choosing the sophomore. Translating this scenario into combinations we get "6 Choose 1". Plugging this into the combinations formula (=Combin(6,1), we get 6. Thus, we have 10 ways to pick the 2 freshmen and 6 ways to choose the sophomore. Finally, to find the total number of combinations, we apply the multiplication rule: 10*6=60, so 60 will be our answer.
Suppose the heights of women on the soccer team are normally distributed with mean of 67 inches and a standard deviation of 5 inches. One woman is chosen at random. Find the probability her height is more than 80 inches.
Since this is a normal distribution problem, we can find the z-score of 80 inches. The formula will be: z=(x-mu)/o Where x is the raw value, mu is the mean, and sigma is the standard deviation In our case, we have: x=80, mu=67 and sigma=5. We plug this in: z=(80-67)/5=13/5=2.6 Once we have the z-score, we can use the normal distribution table to find the area that lies to the left of it. After consulting the table (in the book), looking up the 2.6 row and the .00 column, we get the area of .9953. This will be the area to the left of our z-score. Now, we need to find the area to the right of the z-score, as we want to find the probability of a value of x greater than 80. Since the total area under the normal curve is 1, we subtract: 1-.9953=.0047. This is the area to the right of the z-score corresponding to x=80. Therefore, this will be our probability the height of a randomly chosed woman is greater than 80.
Find the equation of a line perpendicular to y=-3x+2, that goes through the point (0,1).
First, let us determine the slope of the line we are looking for. We know that the slope of a perpendicular line has to be the opposite reciprocal of the current slope. Since the slop eof the given line is -3, the slope of the perpendicular line will be 1/3. Now, we can start writing the equation of the line. We plug in 1/3 for m, into the equation y=mx+b. We get: y=1/3x+b Now, we just make sure the line goes through our point (0,1). For this, we just plug this point into the current equation, getting: 1=1/3*0+b From here, we can solve for b. We get: b=1. Now, we just plug this back into the equation we started. We get: y=1/3x+1, which is now the equation of the line, perpendicular to y=-3x+2, and goes through the point (0,1).