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OleMash [197]
3 years ago
10

Find the number of ways of arranging the numbers

Mathematics
2 answers:
Doss [256]3 years ago
7 0

First of all, note that all integers are either 0,1, or 2 modulo 3 (if you're not familiar with this terminology, it means that every integer is either a multiple of 3, or it is 1 or 2 away from a multiple of 3).

So, we can think of our numbers as

\begin{array}{c|c}x&x\mod 3\\0&0\\1&1\\2&2\\3&0\\4&1\\5&2\\6&0\\7&1\\8&2\\9&0\end{array}

In order to make sure that the sum of any three adjacent numbers is divisible by 3, we have to make sure that any group of 3 three adjacent numbers contains a 0, a 1 and a 2. This is possible only if we arrange our 9 numbers in 3 groups of 3 numbers containing 0,1 and 2 exactly once, repeating always the same pattern.

For example, we could arrange our numbers following the pattern

0,1,2,0,1,2,0,1,2

or

2,0,1,2,0,1,2,0,1

We have 3!=6 possible patterns. Suppose for example that we choose the pattern

0,1,2,0,1,2,0,1,2

One possible way of following this pattern would be the arrangement

3,1,2,6,4,5,9,7,8

In fact, we substituted every '0' with a multiple of 3 (3, 6 or 9), every '1' with a number 1 away from a multiple of 3 (1, 4 or 7) and every '2' with a number 2 away from a multiple of 3 (2, 5 or 8).

This means that, once we fix a patter, we have 3 choices for the first 3 slots, 2 choices for the next 3 slots, and the final slot will be fixed. So, we have

3\cdot 3\cdot 3\cdot 2 \cdot 2 \cdot 2 = 216

possible ways of following a fixed pattern. Since the number of patterns was 6, we have

216\cdot 6 = 1296

possible arrangements.

larisa [96]3 years ago
5 0

Answer: 432

Step-by-step explanation:

Each number is a multiple of 3, or 1 more than a multiple of 3, or one less than a multiple of 3.  These number we will call x's, y's, and z's.

If we add x + y + z, then we get x, because y and z will even each other out, since y is 1 more than multiple of 3 and z is one less.

Because of this, we can conclude that we have to arrange the numbers in a repeating pattern, like x, y, z, x, y, z, ... or z, y, x, z, y, x, ...

In this set 1, 2, 3, ..., 9, there are 3 x's, 3 y's, and 3 z's.

To fill up a pattern, there are 3 choices for the first x, y, z, 2 choices for second, then 1 choice.

3^3 * 2^3 * 1^3 = 27*8*1 = 216

This is equal for both patterns, so there are 216 * 2 = 432 ways to arrange the numbers.

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Answer:

Tina would save the same amount using either option after 20 months.

With either option, Tina would save $700.

Step-by-step explanation:

This problem can be modeled by a first order equation:

Where Tina's saved money after n months is:

S(n) = S(0) + rn, where S(0) is the money put away initially and r is how much she saves every month.

The first option is to put $500 away initially and save $10 every month, so:

S_{1}(n) = 500 + 10n

The second option is to put $100 away initially and save $30 every month, so:

S_{2}(n) = 100 + 30n

After how many months would Tina save the same amount using either option?

It will happen at the month n in which S_{1}(n) = S_{2}(n), so:

S_{1}(n) = S_{2}(n)

500 + 10n = 100 + 30n

500 - 100 = 30 - 10n

400 = 20n

20n = 400

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Tina would save the same amount using either option after 20 months.

How much would she save with either option?

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With either option, Tina would save $700.

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Step-by-step explanation:

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We expect a car's highway gas mileage to be related to its city gas mileage (in miles per gallon, mpg). Data for all 1137 vehicl
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Answer:

a) As we can see on the equation given the slope os 1.033 since is the value next to the city mpg (independent variable)

And on this case since the value for the slope is positive the best interpretation is:

E. On the average, highway mileage increases by 1.033 mpg for each additional mpg in city mileage

Because we see the if we increase on 1 unit the city mpg then the highway mpg would increase 1.033

b) For this case as we can see in the model the free term correspond to 6.785 and that correspond to the intercept for this case.

And the best interpretation is given by:

B. Because this is the highway mileage for zero city mpg.

For this reason is not meaningful since we are not interested on 0 or negative city mpg.

c) highway mpg = 6.785 + (1.033 × 18)=25.38 mpg

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d) See the figure attached.

Step-by-step explanation:

For this case we have the following linear model:

highway mpg = 6.785 + (1.033 × city mpg)

(a) What is the slope of this line? Say in words what the numerical value of the slope tells us.

As we can see on the equation given the slope os 1.033 since is the value next to the city mpg (independent variable)

And on this case since the value for the slope is positive the best interpretation is:

E. On the average, highway mileage increases by 1.033 mpg for each additional mpg in city mileage

Because we see the if we increase on 1 unit the city mpg then the highway mpg would increase 1.033

(b) What is the intercept? (Enter your answer to three decimal places.)mpgExplain why the value of the intercept is not statistically meaningful.

For this case as we can see in the model the free term correspond to 6.785 and that correspond to the intercept for this case.

And the best interpretation is given by:

B. Because this is the highway mileage for zero city mpg.

For this reason is not meaningful since we are not interested on 0 or negative city mpg.

(c) Find the predicted highway mileage for a car that gets 18 miles per gallon in the city. (Round your answer to two decimal places.)______ mpgFind the predicted highway mileage for a car that gets 25 miles per gallon in the city. (Round your answer to two decimal places.)______ mpg

For this case we just need to replace into the formula and we got:

highway mpg = 6.785 + (1.033 × 18)=25.38 mpg

highway mpg = 6.785 + (1.033 × 25)=32.61 mpg

(d) Draw a graph of the regression line for city mileages between 10 and 50 mpg. (Be sure to show the scales for the x and y axes.)

See the figure attached

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