7. Eleven students go to lunch. There are two circular tables in the dining hall, one can seat 7 people, the other can hold 4. I
1 answer:
Answer:
239,580 ways of seating
Step-by-step explanation:
11 students will be divided into 2 groups. One group of 7 people and one group of 4 people. So first we need to find the number of ways of dividing 11 students into these 2 groups.
First group is of 7 people. We have to select 7 people out of 11. The order of selection does not matter so this is a combination problem. Selecting 7 people from 11 can be expressed as 11C7.
Formula for combination is:
For the given case this would be:
So, there are 330 ways of selecting a group of 7 from 11 students. When these 7 students are selected the remaining 4 will go to the other group. So, we can say there are 330 ways to divide the 11 students in groups of 7 and 4. Note that if you start with group of 4 students, the answer will still the same because 11C4 is also equal to 330.
Next we have to arrange 7 students on a round table. The number of possible arrangements would be = (7 - 1)! = 6! = 720
Similarly, to arrange 4 people on a round table, the number of possible arrangements would be = (4 - 1)! = 3! = 6
Since, for each selection of the 330 groups, there are 720 + 6 possible seating arrangements, so the total number of possible seating arrangements would be:
330 ( 720 + 6) = 239,580 ways
Thus, there are 239,580 ways of seating 11 students.
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Answer: Last Option
Step-by-step explanation:
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Answer:
Step-by-step explanation:
The price that will maximize sales and the number of projected sales can be found in the vertex of a quadratic equation. We need to use the puzzle of the info given to somehow create a set of biomials that can be multiplied together to get this quadratic. We will separate the number of books sold from the price of the books in a table of sorts:
# of books $ per book
We know that a number of 825 books was sold when the price per book was $26.
# of books $ per book
825 $26
If we have a $1 decrease in price per book, let's use as our unknown the number of $1 decreases in price per book. In other words, x = # of $1 decreases.
If we decrease the price per book by $1, we are decreasing the price by 1x which is 1 dollar.
If, when we decrease the price per book by $1, we are selling the original 825 books plus another 75 books at the $1 decrease per book. Now we have in our table
# of books $ per book
825 + 75x 26 - x
Under # of books, the expression in words says, "we are still selling 825 books, but now we are adding an additional 75 books per month when we lower the cost $1 per book".
Under $ per book, the expression in words says, "the cost per book is the starting cost minus $1".
To get the quadratic that results from this, multiply the 2 expressions together to get
or, if we want to keep things positive:
We can factor a 75 out of each term to make the numbers a bit smaller:
Use the quick formulas for h and k to solve for the vertex. You could complete the square to get there, but once you know these formulas, there's no need to go through the very long long process of completing the square.
and
These formulas come from the quadratic equation. h here will give us the number of $1 decreases we need to maximize k, the profit from this amount decreases.
Our a = 1, b = -15 and c = -286. Filling in for h:
and
so
h = 7.5 That is the number of decreases
Filling in for k:
and
so
k = 342.25 That is the maximum projected sales from this amount of decreases in price.
But we are not done. We need to use the 7.5 to find out what the best price would be to maximize the sales and therefore, the profit. Going back to our cost expression, 26 - x, using 7.5 for x:
26 - 7.5 = $18.50
This means that if we sell books at $18.50 rather than $26, we can expect to earn $342.25 per month in book sales.